# Terahertz Scale Micro-Bunching Instability Driven by NEG-Coating Resistive-
Wall Impedance

Weiwei Li  Tianlong He<EMAIL_ADDRESS>Zhenghe Bai  National Synchrotron
Radiation Laboratory, University of Science and Technology of China, Hefei,
Anhui, 230029, China

###### Abstract

Non-evaporable getter (NEG) coating is widely required in the next generation
of light sources and circular $e^{+}e^{-}$ colliders for small vacuum pipes to
improve the vacuum level, which, however, also enhances the high-frequency
resistive-wall impedance and often generates a resonator-like peak in the
terahertz frequency region. In this paper, we will use the parameters of the
planned Hefei Advanced Light Facility (HALF) storage ring to study the impact
of NEG coating resistive-wall impedance on the longitudinal microwave
instability via particle tracking simulation. Using different NEG coating
parameters (resistivity and thickness) as examples, we find that the impedance
with a narrow and strong peak in the high frequency region can cause micro-
bunching instability, which has a low instability threshold current and
contributes to a large energy spread widening above the threshold. In order to
obtain a convergent simulation of the beam dynamics, one must properly resolve
such a peak. The coating with a lower resistivity has a much less sharp peak
in its impedance spectrum, which is helpful to suppress the micro-bunching
instability and in return contributes to a weaker microwave instability.

PACS numbers

29.27.Bd, 41.75.Ht

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The non-evaporable getter (NEG) coating [1] has been successfully applied to
inner surfaces of many vacuum chambers of particle accelerators. It can
provide distributed pumping along vacuum chambers, thus, the specified
ultrahigh vacuum pressure level could be met with a reduced number and a size
of external pumps.

The resistivity wall (RW) impedance, produced by the finite conductivity of
the beam vacuum chamber, plays an important, often dominant, role in modern
accelerators, especially in those with a small transverse size of the vacuum
chamber [2, 3]. The presence of the NEG coating films makes the surface
resistance of the beam pipe higher than the one without coating and the
resistive wall effect is more pronounced. This impact is especially important
for large machines with small beam pipe dimensions such as circular
$e^{+}e^{-}$ colliders [4, 5] and diffraction-limited storage rings (DLSRs)
[6, 7, 8]. In addition, there could be also an uncertainty on the coating
conductivity measurements in the high frequency region which may give
inaccurate predicts about the instability threshold. In order to reduce the RW
impedance contribution and the uncertainty due to the mostly unknown coating
resistivity, one of the best ways is to reduce the NEG coating thickness.
Several DLSR projects [6, 7, 8] have set the NEG coating thickness targets
equal to or less than 1 um. Recent study [8] also shows that there is a regime
in which a coating with a lower resistivity produces even a larger loss factor
than that with a higher resistivity.

Table 1: Main Parameters of the Vacuum Chambers Type | Material + film (thickness: $\mathrm{\SIUnitSymbolMicro m}$) | Shape | Aperture/Radii (mm) | Length (m)
---|---|---|---|---
Main Chamber | CuCrZr+NEG (d) | Round | 13 | 344.2
Fast Corrector | Inconel + NEG (d) | Round | 13 | 7.2
beam pipes with antechamber | Stainless steel + Cu (20) | Round | 13 | 48.2
out vacuum Insertion devices | Al | Elliptical | $8\times 26$ | 43
In-Vacuum Undulators | NdFeB + Ni (75) + Cu (75) | Rectangular | $6\times 65$ | 5.4
Others (ie., bellows, flanges) | Stainless Steel | Round | 13 | 32

The RW has a strong longitudinal impedance in the high frequency region,
contributing to a sharp variation of the point-charge longitudinal wakefield
in very short distances. The presence of the NEG coating will further enhance
the high-frequency impedance and often generates a resonator-like peak [9,
10]. The multi-particle tracking simulations [11, 12, 13, 14] are widely used
to study the beam dynamics but may have computational issues to study the
longitudinal microwave instability (MWI), where a large number of simulation
particles is needed to study the response of small-scale bunch structures to
high frequency wakefield components. If not equipped with suitable algorithms
(i.e., smoothing/filtering techniques, fine grids, etc.), the simulation can
fail to produce reliable results [15]. Since the collective behavior usually
doesn’t depend on the behavior of the wakefield/impedance at very small length
scales/very high frequencies, one popular solution is to use the wake
potential of a very short Gaussian bunch of rms length $\overline{\sigma}_{s}$
(sometimes called pseudo-Green function) in place of the point-charge
wakefield [3, 4, 16, 17]. Then the impedance used in tracking simulation
becomes that of the point-charge multiplied by a Gaussian filter. The
remaining item is to determine the required length $\overline{\sigma}_{s}$,
which effectively means finding the frequency range over which the impedance
affects the dynamics. Usually the guideline that is 10 or 15 times smaller
than that of the equilibrium beam gives a reasonable estimate [3] . However,
the MWI simulations are quite sensitive to numerical noise, so it is necessary
to vary different tracking parameters to make sure that the tracking results
are accurate.

In this paper, the Hefei Advanced Light Facility (HALF) [18], a fourth
generation light source in design, will be used to study the impact of the NEG
coating parameters. The equilibrium beam rms length at zero current limit is
2.1 mm, however, in order to obtain a quasi convergent simulation of the beam
dynamics, the required length $\overline{\sigma}_{s}$ should typically be even
smaller than 0.02 mm and millions of marco-particles are necessary. Under
convergent simulations, we find that the coating with a high resistivity (in
the order of $10^{-5}~{}$\Omega~{}\mathrm{m}$$), whose impedance has a sharp
peak in the terahertz region, can cause an undesirable micro-bunching
instability (MBI), with a low threshold and a large energy spread widening.

This paper is organized as follows. In Sec. II, the RW impedance with
different NEG coating parameters of HALF will be presented. In Sec. III, the
particle tracking method will be introduced. Section IV shows the simulation
results with different tracking and coating parameters. The conclusions and
discussions are presented in Sec. V.

##  II  resistive wall impedance in half

We consider a simplified model of the ring consisting 6 parts and the main
parameters of the vacuum chambers are listed in Table 1. The resistivities of
the materials are listed in Table 2.

Table 2: Main parameters of HALF Material | Resistivity ($\Omega~{}\mathrm{m}$)
---|---
Cu | $1.68\times 10^{-8}$
CuCrZr | $2.3\times 10^{-8}$
Al6063 | $3.16\times 10^{-8}$
Ni | $6.93\times 10^{-7}$
SS316L | $7.41\times 10^{-7}$
Inconel 625 | $1.29\times 10^{-6}$

The NEG resistivity value depends on the compound composition and coating
method [19]. The resistivity measurement results also have very large
discrepancy using different methods [20]. Thus three different resistivity
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}$ values will be taken into account:
$1\times 10^{-5}~{}$\Omega~{}\mathrm{m}$$, $5\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$. The target of NEG coating thickness is
$d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ for the HALF project, but two
different film thicknesses will be studied for comparison: 0.5
$\mathrm{\SIUnitSymbolMicro m}$ and 1 $\mathrm{\SIUnitSymbolMicro m}$.

The resistive wall impedance is computed using the ImpedanceWake2D (IW2D) code
[21], which solves for a circular geometry and then applies a Yokoya factor
for elliptical or rectangular cases [22, 23]. Their longitudinal impedance as
a function of frequency is shown in Fig. 1. At low frequency, all the
impedance is similar since the thickness of coating is much smaller than its
skin depth. At high frequency, the impedance is greatly enhanced with NEG
coatings and generates a resonator-like peak in the terahertz region. If the
coating resistivity or thickness is smaller, the impedance will be more close
to that of no coating. A smaller coating resistivity will reduce the quality
factor and the peak impedance. A thinner coating thickness will make the peak
frequency shift downward.

Figure 1: The real (top) and imagine (bottom) parts of the longitudinal
impedance for different coating parameters.

It is useful to define the effective impedance as [24, 25, 26]

${\left({\frac{{{Z_{\parallel}}}}{n}}\right)_{{\rm{eff}}}}=\frac{{\int_{-\infty}^{\infty}{{Z_{\parallel}}\left(\omega\right)\frac{{{\omega_{0}}}}{\omega}h\left(\omega\right)d\omega}}}{{\int_{-\infty}^{\infty}{h\left(\omega\right)d\omega}}},$
(1)

where $n=\omega/\omega_{0}$ is the revolution harmonic number, $\omega_{0}$ is
the revoltion angular frequency,
$h\left(\omega\right)=\tilde{\lambda}\left(\omega\right){{\tilde{\lambda}}^{*}}\left(\omega\right)$
is the bunch power spectrum, $\tilde{\lambda}\left(\omega\right)$ is the
Fourier transform of the longitudinal charge density $\lambda\left(t\right)$.
Assuming a Gaussian bunch,
$h\left(\omega\right)=e^{-\omega^{2}\sigma_{t}^{2}}$, where $\sigma_{t}$ is
the rms bunch length in time. The effective impedance with natural bunch
length (7 ps) for different NEG coating parameters are listed in Table 3.

Table 3: The effective impedance $($\mathrm{m}\Omega$)$ with natural bunch length for different NEG coating parameters d ($\mathrm{\SIUnitSymbolMicro m}$) $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}($\Omega~{}\mathrm{m}$)$ | $1\times 10^{-5}$ | $5\times 10^{-6}$ | $1\times 10^{-6}$
---|---|---|---
1 | 77.7 | 77.5 | 76.5
0.5 | 67.2 | 67.1 | 66.6

##  III  Multi-particle Tracking Simulation

Most of the macroparticle tracking codes compute the wake potential as the
convolution between the longitudinal wake function $w_{\parallel}(t)$, i.e.,
the Green function of a point charge, and the bunch distribution
$\lambda\left(t\right)$: [12]

${W_{\parallel}}\left(t\right)=\int_{-\infty}^{t}{{w_{\parallel}}\left({t-t^{\prime}}\right)\lambda\left({t^{\prime}}\right)}dt^{\prime}.$
(2)

The longitudinal wake function can be expressed in terms of the longitudinal
impedance by an inverse Fourier transform

${w_{\parallel}}\left(t\right)=\frac{1}{{2\pi}}\int_{-\infty}^{\infty}{{Z_{\parallel}}\left(\omega\right){e^{i\omega
t}}}d\omega.$ (3)

For the convenience of numerical calculation, the time coordinate in Eq. 2
should be equi-spaced and discrete Fourier Transforms can be used. The bin
size is $\Delta_{t}=\frac{0.5}{F_{m}}$, where $F_{m}$ is the maximum frequency
of the impedance in Eq. 3. $\lambda\left(t\right)$ is obtained by counting the
particle number in each bin and $>\sim$ 1000 particles per bin is typically
required. However, the input RW wake function can cause some problems to
simulations based on this approach, since it covers a very high frequency
range and a very large number of slices would be necessary, thus increasing
the computational load. As discussed in Sec. I, the pseudo-Green function from
a very short Gaussian pulse will be used. The pulse length
$\overline{\sigma}_{s}$ will determine the frequency reach of the impedance
calculation and needs to be much smaller than the real bunch length used in
tracking simulations to cover the spectrum of interest. However, in order to
resolve the impedance peak, the computational load is still very heavy.

The STABLE code [27] will be dedicated to conduct multi-particle tracking
simulations for longitudinal beam dynamics studies. It is implemented in a
MATLAB environment with the usage of the state-of-the-art of graphics-
processing-unit acceleration technique so that the tracking efficiency is
significantly improved. The original version of STABLE is written for multi-
bunch and multi particle simulation, and a 2D matrix is used to store the
macro-particle’s coordinates, with each column corresponding to one bunch. In
order to accurately simulate the single bunch dynamics, which usually requires
millions or even tens of millions of macro-particles, we need only modify the
STABLE code by dividing the macro-particles of single bunch into multiple
parts and storing them in each column of the 2D matrix. We can separately
count the bin-distribution of each column, and then summarize them to obtain
the total bunch distribution. In addition, a fixed bin width instead of the
number of bins is set in default. Therefore, the bin number will increase as
the bunch lengthening. The remaining operations, such as convolution of the
bunch distribution and the short-range wake (or short-bunch wakepotential),
and interpolation to obtain the short-range wake kick of each macro-particle,
can be same as those in the original version of STABLE.

##  IV  NUMERICAL RESULTS FOR THE PARAMETERS OF HALF

In this section, the impact of NEG-coated resistive-wall impedance on the
longitudinal beam dynamics is investigated by tracking simulations in the
framework of HALF project. The main parameters of the HALF storage ring with
insertion devices are summarized in Table 4.

Table 4: Main parameters of HALF Parameter | Symbol | Value
---|---|---
Ring circumference | C | $480\text{\,}\mathrm{m}$
Beam energy | $E_{0}$ | $2.2\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$
Nominal beam current | $I_{0}$ | $350\text{\,}\mathrm{m}\mathrm{A}$
Longitudinal damping time | $\tau_{z}$ | $14\text{\,}\mathrm{m}\mathrm{s}$
Momentum compaction | $\alpha_{c}$ | $9.4\times 10^{-5}$
Natural energy spread | $\sigma_{\delta}$ | $7.3\times 10^{-4}$
Harmonic number | $h$ | $800\text{\,}$
Energy loss per turn | $U_{0}$ | $400\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
Voltage of MC | $V_{RF}$ | $1.2\text{\,}\mathrm{M}\mathrm{V}$
Natural rms bunch length | $\sigma_{t0}$ | $7\text{\,}\mathrm{p}\mathrm{s}$

###  IV.1 Convergence Study

We first carry out the convergence study with two typical examples, the
coatings with $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ and
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$, $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
respectively.

####  IV.1.1 Pseudo-Green Function

The wake potentials for the short Gaussian bunch of $\overline{\sigma}_{s}=$
0.1 mm and 0.01 mm are shown in Fig. 2, where the positive (negative) value
means energy loss (gain), and they will be used instead of the wake function
generated from a point charge. To calculate the effective impedance or loss
factors for a perfect Gaussian bunch distribution with nominal bunch length
$\sigma_{t0}$ from the pseudo-Green function, using the wake potentials of
$\overline{\sigma}_{s}=$ 0.1 mm can usually obtain sufficiently accurate
results. Note that $\overline{\sigma}_{s}=$ 0.1 mm is more than 20 times
shorter than the natural bunch length $\sigma_{t0}$.

The longitudinal impedance multiplied by the Fourier spectrum
$\tilde{\lambda}\left(\omega\right)$ of the Gaussian bunches with different
$\overline{\sigma}_{s}$ is plotted in Fig. 3. In order to resolve the
impedance peak clearly, $\overline{\sigma}_{s}$ should be smaller than 0.02
mm.

Figure 2: Longitudinal wake potentials for different
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}$ and $\overline{\sigma}_{s}$. The
coating thickness is $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$.

Figure 3:  The real part (solid) and the absolute value of the imagine part
(dashed) of the longitudinal impedance multiplied by different Gaussian
filters for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ (top) and
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ (bottom).The coating thickness is
$d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$.

####  IV.1.2 The Case for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$

Figure 4 shows the predicted energy spread and bunch lengthening from the
tracking simulations as a function of the single bunch current for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with various values of $\overline{\sigma}_{s}$, where the particle number
$N_{p}$ is 5 millions (M), 40,000 turns are tracked for each current and the
bin size $\Delta_{t}$ is set to be 0.02 ps. The corresponding $F_{m}$ is 25
THz, which should be high enough to cover the frequency region of interest.
Another guideline is that $\Delta_{t}$ should be small enough to resolve the
wake potential generated from the short Gaussian bunch with
$\overline{\sigma}_{s}$. The mean value and standard deviation of bunch length
and relative energy spread are computed on last 10,000 turns. To benchmark the
simulation results from the STBALE code, the Pelegant code [28] is also used
with the same parameters except for fewer current and $\overline{\sigma}_{s}$
values and the results are also marked in Fig. 4. In order to obtain one data
point in Fig. 4, it takes about 200 min for Pelegant using 80 CPU cores, while
less than 10 min for STABLE using 3584 CUDA cores. Good agreements are
achieved since their underlying physical models are the same. Thus we will
only use the STABLE code for the other simulations.

Figure 4:  The rms energy spread (top) and bunch length (bottom) versus single
bunch current for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with different $\overline{\sigma}_{s}$. The solid lines are the mean values
and the dashed lines including their fill areas represent the standard
deviation obtained by STABLE with current step of 0.05 mA. The discrete error
bars are obtained by Pelegant with current step of 0.5 mA.

To validate the choice of the bin size $\Delta_{t}$ and the particle number
$N_{p}$, convergence studies are performed for the the case of
$\overline{\sigma}_{s}=$ 0.01 mm, since a shorter $\overline{\sigma}_{s}$
requires more severe convergence conditions. The corresponding energy spreads
are shown in Fig. 5. There is no significant variation when $\Delta_{t}$
decreases from 0.02 ps to 0.01 ps or $N_{p}$ varies from 2 M to 10 M.

Figure 5: The energy spread versus single bunch current for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
$\overline{\sigma}_{s}$=0.01 mm with different $N_{p}$ and $\Delta_{t}$. The
lines represent the mean values and the error bars represent the standard
deviation.

As seen in Fig. 4, at the low current below the MWI threshold, the simulations
using a relative long $\overline{\sigma}_{s}=$ 0.01 mm have already given
enough convergent results. However, in order to accurately evaluate the MWI,
one must properly resolve the resonator-like peak impedance. To obtain a full
convergent simulation for the coating with
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$,
the required $\overline{\sigma}_{s}$ is 0.02 mm. The MWI behavior can be
significantly underestimated if the wakefield resolution
$\overline{\sigma}_{s}$ is not sufficient.

####  IV.1.3 The Case for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$

Figure 6: The rms energy spread versus single bunch current for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with different $N_{p}$ and $\overline{\sigma}_{s}$. The lines represent the
mean values and the error bars represent the standard deviation.

Figure 6 shows the predicted energy spread from the tracking simulations as a
function of the single bunch current for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with various values of $\overline{\sigma}_{s}$ and $N_{p}$, where
$\Delta_{t}=$ 0.02 ps and the current step is 0.25 mA. For
$\overline{\sigma}_{s}=$ 0.1 mm, if $N_{p}=$ 2 M is adopted, it shows obvious
energy spread widening, but as with the increment of $N_{p}$, the energy
spread widening becomes smaller and when $N_{p}$ increases to 50 M, there is
nearly no MWI within 2 mA. However, the peak impedance is still not resolved,
thus we further study the case of $\overline{\sigma}_{s}=$ 0.01 mm and 0.005
mm. With the same $N_{p}$, their results are close, so
$\overline{\sigma}_{s}=$ 0.01 mm should be enough to cover the frequency
region of interest. Within 1.5 mA, the energy spread widening becomes smaller
as with the increment of $N_{p}$, but the full convergence is still not
achieved even when $N_{p}=$ 50 M, but the energy spread widening becomes
relative small. Therefore we can conclude that there is no or very weak MWI
within 1.5 mA and it is reasonable to use a long $\overline{\sigma}_{s}$ such
as 0.1 mm to filter out the high frequency wakefield components. For
$\overline{\sigma}_{s}=$ 0.01 mm or 0.005 mm at a high current of 2 mA, there
is no significant variation when $N_{p}$ varies from 10 M to 50 M and obvious
energy spread widening can be seen, so in this situation one should also use a
small $\overline{\sigma}_{s}$ to resolve the resonator-like peak impedance in
order to accurately predict the MWI behavior.

###  IV.2 Micro-bunching Instability Phenomena

In the previous subsection, it has shown that the coating with
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
causes a much more serious MWI than that with
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
although they are close in effective impedance with natural bunch length, so
there is a practical interest in exploring the underlying mechanism.

Figure 7: The rms energy spread evolution over the pass turn for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
at 0.5 mA together with the longitudinal phase space plots ($t-\delta$) at
three marked points.

The energy spread evolution over the pass turn for
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with $\overline{\sigma}_{s}=$ 0.01 mm and $N_{p}=$ 20 M at current of 0.5 mA
together with the longitudinal phase space distributions at three different
turns are shown in Fig. 7. More simulated particles are used just to make the
plots of the phase spaces more smooth and clear. There appears strong
sawtooth-shaped fluctuations of energy spread over the pass turn. The error
bars in Figs. 4 and 5 also characterize the amplitudes of the fluctuations.
The MBI in the phase space is visible corresponding to a modulation frequency
around 0.58 THz. It implies that the sharp peak in the impedance spectrum
plays an important role in the MBI. A possible reason is that when the peak is
narrowband (or has a high quality factor), the corresponding wakefield lasts
for several oscillation cycles (as shown in Fig. 2), which allows the
wakefield from the micro-bunches far apart to be coherently enhanced. While
for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$,
the peak is more broadband, the wakefield attenuates quickly with the
increasing of distance, which prevents the cooperation between the micro-
bunching fluctuations far apart.

###  IV.3 Impact of Coating Parameters

We have shown the coating with
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
can cause the MBI effect with a low current threshold for the HALF ring. To
avoid its occurrence, there is a practical interest in exploring the
dependence on the coating parameters.

Figure 8: The rms energy spread (top) and bunch length (bottom) versus single
bunch current for different coatings. The solid lines are the mean values and
the dashed lines including their fill areas represent the standard deviation.
The discrete error bars are obtained using $N_{p}=$ 20 M to validate the
convergences.

We consider the coating parameters given in Sec. II, in which their impedances
are given. Figure 8 shows the predicted energy spread and bunch lengthening
from the tracking simulations as a function of the single bunch current for
different coating parameters, where the current step is 0.25 mA, $\Delta_{t}=$
0.02 ps, $\overline{\sigma}_{s}=$ 0.01 mm which is small enough to resolve the
resonator-like peak impedance, and the particle number is 10 M except for the
case of $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
where 50 M particles are used. The convergence studies for the cases
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$, $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ and
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$, $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ have
been done in the subsection IV.1. To validate the choice of $N_{p}$ for the
other cases, we also carry out the simulations using $N_{p}=$ 20 M and the
results are also plotted in Fig. 8 as error bars, and there is no significant
variation for each case. With the same coating thickness
$d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ or $0.5~{}$\mathrm{\SIUnitSymbolMicro
m}$$, the MWI for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=5\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ is less serious and has a higher threshold
current than that for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$, but there still exits MBI when the current
exceeds the threshold. For the coating with resistivity
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ or $5\times 10^{-6}~{}$\Omega~{}\mathrm{m}$$,
reducing the thickness from $1~{}$\mathrm{\SIUnitSymbolMicro m}$$ down to
$0.5~{}$\mathrm{\SIUnitSymbolMicro m}$$ is helpful to weaken the MBI, since
the peak frequency in the impedance spectrum for
$d=0.5~{}$\mathrm{\SIUnitSymbolMicro m}$$ is higher than that for
$d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$ as seen in Fig. 1. For the coating
with $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$,
it doesn’t contribute to MBI, so it has a much higher instability threshold
and smaller energy spread widening. For the coating with
$\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-6}~{}$\Omega~{}\mathrm{m}$$, reducing the coating thickness from
$1~{}$\mathrm{\SIUnitSymbolMicro m}$$ down to
$0.5~{}$\mathrm{\SIUnitSymbolMicro m}$$ will make a lower instability
threshold, this is because the impedance peak of the latter is much sharper as
shown in Fig. 1, which can also lead to MBI.

###  IV.4 Impact of Bunch Lengthening with HHC

Bunch lengthening with higher harmonic cavities (HHCs) is also a very helpful
means to fight against most collective effects, including the MWI. A passive
superconducting 3rd harmonic cavity will be installed in the HALF storage ring
[29, 30]. Instead of multibunch simulations, we just carry out single bunch
simulations by introducing an ideal HHC voltage potential because of the heavy
computational loads and making the bunch length increase to a factor of 2 or 3
at zero current. Figure 9 shows the predicted energy spread and bunch
lengthening from the tracking simulations as a function of the single bunch
current for the case of $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with different bunch lengthening factors. In order to obtain a convergent
simulation, $\overline{\sigma}_{s}$ should still be as small as that without
HHC to resolve the resonator-like peak impedance. The bunch lengthening with
HHC can raise the MWI threshold since it lowers the charge density.

Figure 9: The rms energy spread (top) and bunch length (bottom) versus single
bunch current for $\rho_{$\mathrm{N}\mathrm{E}\mathrm{G}$}=1\times
10^{-5}~{}$\Omega~{}\mathrm{m}$$ and $d=1~{}$\mathrm{\SIUnitSymbolMicro m}$$
with different bunch lengthening factors (LFs). The solid lines are the mean
values and the dashed lines including their fill areas represent the standard
deviation.

##  V  Conclusion and Discussion

In this paper, we have studied the impact of NEG coating resistive-wall
impedance on the longitudinal microwave instability (MWI) for the HALF storage
ring via particle tracking simulation, where the wake potential of a very
short Gaussian bunch with rms length of $\overline{\sigma}_{s}$ serves as
pseudo Green function. In order to obtain a quasi convergent simulation of the
beam dynamics, $\overline{\sigma}_{s}$ should be small sufficiently to resolve
the peak impedance in the high frequency region. For the cases presented in
this paper, $\overline{\sigma}_{s}$ should be at most 0.02 mm, which is more
than 100 times shorter than that of the equilibrium beam at zero limit.
Otherwise, one is likely to underestimate the MWI behavior. Recent microwave
instability studies [17] for the ESRF-EBS show that the measured current
threshold is significantly lower than the stimulated one, but all the
wakefield models are computed with $\overline{\sigma}_{s}$ =1 mm, which is far
from satisfying the resolution of the high frequency RW components and can be
a possible answer to explain the discrepancy.

The effective impedance is often used to evaluate the MWI, however, our
studies show that the characteristics of the peak in the high frequency region
are also critical. A strong and narrowband peak can cause an undesirable
micro-bunching instability (MBI), which has a low threshold current and makes
the dynamics of MWI more complex. For a high coating resistivity (in the order
of $10^{-5}~{}$\Omega~{}\mathrm{m}$$), reducing the thickness is helpful to
weaken the MBI by shifting the peak impedance to higher frequency but the MBI
is still dangerous. Another effective way to suppress the MBI is to reduce the
coating resistivity, which has a broadband impedance. The bunch lengthening
with HHC can raise the MWI/MBI threshold. A storage ring with NEG coating of
low resistivity applied to inner surfaces of many vacuum chambers and a low
frequency main cavity [31, 32] can still suffer from the MBI since it has high
single bunch charges.

The NEG-coating resistive-wall high frequency impedance can play an important
role in the MBI, so accurate measurements of the NEG resistivity are very
important in order to perform accurate simulations on their impact on the beam
dynamics. The MBI will reduce the beam quality but also has the potential to
tailor the emitted CSR radiation and its fluctuations for possible
applications of the terahertz radiation.

###### Acknowledgements.

The authors would like to thank Sihui Wang at USTC and Na Wang at IHEP for
useful discussions on NEG coatings and Biaobin Li at USTC for useful
discussions on the numerical convergence. This work was supported by National
Natural Science Foundation of China (No. 12105284 and No. 11875259) and the
Fundamental Research Funds for the Central Universities (No. WK2310000090).

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# From (secure) $w$-domination in graphs to protection of lexicographic
product graphs

A. Cabrera Martínez, A. Estrada-Moreno, J. A. Rodríguez-Velázquez
Universitat Rovira i Virgili
Departament d’Enginyeria Informàtica i Matemàtiques
Av. Països Catalans 26, 43007 Tarragona, Spain.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Let $w=(w_{0},w_{1},\dots,w_{l})$ be a vector of nonnegative integers such
that $w_{0}\geq 1$. Let $G$ be a graph and $N(v)$ the open neighbourhood of
$v\in V(G)$. We say that a function $f:V(G)\longrightarrow\\{0,1,\dots,l\\}$
is a $w$-dominating function if $f(N(v))=\sum_{u\in N(v)}f(u)\geq w_{i}$ for
every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be
$\omega(f)=\sum_{v\in V(G)}f(v)$. Given a $w$-dominating function $f$ and any
pair of adjacent vertices $v,u\in V(G)$ with $f(v)=0$ and $f(u)>0$, the
function $f_{u\rightarrow v}$ is defined by $f_{u\rightarrow v}(v)=1$,
$f_{u\rightarrow v}(u)=f(u)-1$ and $f_{u\rightarrow v}(x)=f(x)$ for every
$x\in V(G)\setminus\\{u,v\\}$. We say that a $w$-dominating function $f$ is a
secure $w$-dominating function if for every $v$ with $f(v)=0$, there exists
$u\in N(v)$ such that $f(u)>0$ and $f_{u\rightarrow v}$ is a $w$-dominating
function as well. The (secure) $w$-domination number of $G$, denoted by
($\gamma_{w}^{s}(G)$) $\gamma_{w}(G)$, is defined as the minimum weight among
all (secure) $w$-dominating functions.
In this paper, we show how the secure (total) domination number and the
(total) weak Roman domination number of lexicographic product graphs $G\circ
H$ are related to $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$. For the case of the
secure domination number and the weak Roman domination number, the decision on
whether $w$ takes specific components will depend on the value of
$\gamma_{(1,0)}^{s}(H)$, while in the case of the total version of these
parameters, the decision will depend on the value of $\gamma_{(1,1)}^{s}(H)$.

Keywords: Secure $w$-domination, $w$-domination, weak Roman domination, secure
domination, lexicographic product.

MSC2020: 05C69, 05C76

##  1 Introduction

As usual, $\mathbb{Z}^{+}=\\{1,2,3,\dots\\}$ and
$\mathbb{N}=\mathbb{Z}^{+}\cup\\{0\\}$ denote the sets of positive and
nonnegative integers, respectively. Let $G$ be a graph, $l\in\mathbb{Z}^{+}$
an integer, and $f:V(G)\longrightarrow\\{0,\dots,l\\}$ a function. Let
$V_{i}=\\{v\in V(G):\;f(v)=i\\}$ for every $i\in\\{0,\dots,l\\}$. We will
identify $f$ with the subsets $V_{0},\dots,V_{l}$ associated with it, and so
we will use the unified notation $f(V_{0},\dots,V_{l})$ for the function and
these associated subsets. The _weight_ of $f$ is defined as

$\omega(f)=f(V(G))=\sum_{i=1}^{l}i|V_{i}|.$

Let $w=(w_{0},\dots,w_{l})\in\mathbb{Z}^{+}\times\mathbb{N}^{l}$ such that
$w_{0}\geq 1$. As defined in [4], a function $f(V_{0},\dots,V_{l})$ is a
$w$-_dominating function_ if $f(N(v))\geq w_{i}$ for every $v\in V_{i}$. The
$w$-_domination number_ of $G$, denoted by $\gamma_{w}(G)$, is the minimum
weight among all $w$-dominating functions. For simplicity, a $w$-dominating
function $f$ of weight $\omega(f)=\gamma_{w}(G)$ will be called a
$\gamma_{w}(G)$-function. For fundamental results on the $w$-domination number
of a graph, we refer the interested readers to [4]; the paper where the theory
of $w$-domination in graphs was introduced.

For any function $f(V_{0},\dots,V_{l})$ and any pair of adjacent vertices
$v\in V_{0}$ and $u\in V(G)\setminus V_{0}$, the function $f_{u\rightarrow v}$
is defined by $f_{u\rightarrow v}(v)=1$, $f_{u\rightarrow v}(u)=f(u)-1$ and
$f_{u\rightarrow v}(x)=f(x)$ whenever $x\in V(G)\setminus\\{u,v\\}$.

The authors of this paper [5] introduced the approach of secure $w$-domination
as follows. A $w$-dominating function $f(V_{0},\dots,V_{l})$ is a _secure_
$w$-_dominating function_ if for every $v\in V_{0}$ there exists $u\in
N(v)\setminus V_{0}$ such that $f_{u\rightarrow v}$ is a $w$-dominating
function as well. The _secure_ $w$-_domination number_ of $G$, denoted by
$\gamma_{w}^{s}(G)$, is the minimum weight among all secure $w$-dominating
functions. For simplicity, a secure $w$-dominating function $f$ of weight
$\omega(f)=\gamma_{w}^{s}(G)$ will be called a $\gamma_{w}^{s}(G)$-function.
This approach to the theory of secure domination covers the different versions
of secure domination known so far. For instance, we would emphasize the
following cases of known parameters that we define here in terms of secure
$w$-domination.

  * •

The _secure domination number_ of $G$ is defined to be
$\gamma_{s}(G)=\gamma_{(1,0)}^{s}(G)$. In this case, for any secure
$(1,0)$-dominating function $f(V_{0},V_{1})$, the set $V_{1}$ is known as a
_secure dominating set_. This concept was introduced by Cockayne et al. in
[12] and studied further in several papers, including among others, [2, 3, 10,
11, 19, 21].

  * •

The _secure total domination number_ of a graph $G$ of minimum degree at least
one is defined to be $\gamma_{st}(G)=\gamma_{(1,1)}^{s}(G)$. In this case, for
any secure $(1,1)$-dominating function $f(V_{0},V_{1})$, the set $V_{1}$ is
known as a _secure total dominating set_ of $G$. This concept was introduced
by Benecke et al. in [1] and studied further in several papers, including
among others, [7, 8, 14, 19, 20].

  * •

The _weak Roman domination number_ of a graph $G$ is defined to be
$\gamma_{r}(G)=\gamma_{(1,0,0)}^{s}(G)$. This concept was introduced by
Henning and Hedetniemi [17] and studied further in several papers, including
among others, [6, 10, 11, 22].

  * •

The _total weak Roman domination number_ of a graph $G$ of minimum degree at
least one is defined to be $\gamma_{tr}(G)=\gamma_{(1,1,1)}^{s}(G)$. This
concept was introduced by Cabrera et al. in [8] and studied further in [9].

  * •

The _secure Italian domination number_ of $G$ is defined to be
$\gamma_{{}_{I}}^{s}(G)=\gamma_{(2,0,0)}^{s}(G)$. This parameter was
introduced by Dettlaff et al. in [13].

In this paper we show how the secure (total) domination number and the (total)
weak Roman domination number of lexicographic product graphs $G\circ H$ are
related to $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$. For the case of the secure
domination number and the weak Roman domination number, the decision on
whether $w$ takes specific components will depend on the value of
$\gamma_{(1,0)}^{s}(H)$, while in the case of the total version of these
parameters, the decision will depend on the value of $\gamma_{(1,1)}^{s}(H)$.

We assume that the reader is familiar with the basic concepts, notation and
terminology of domination in graph. If this is not the case, we suggest the
textbooks [15, 16]. For the remainder of the paper, definitions will be
introduced whenever a concept is needed.

##  2 Some tools

Given a $w$-dominating function $f(V_{0},\dots,V_{l})$ and $v\in V_{0}$, we
define

$M_{f}(v)=\\{u\in V(G)\setminus V_{0}:\,f_{u\rightarrow v}\text{ is a
}w\text{-dominating function}\\}.$

Obviously, if $f$ is a secure $w$-dominating function, then
$M_{f}(v)\neq\varnothing$ for every $v\in V_{0}$.

######  Theorem 2.1.

[5] Let $G$ be a graph of minimum degree $\delta$, and let
$w=(w_{0},\dots,w_{l})\in\mathbb{Z}^{+}\times\mathbb{N}^{l}$ such that
$w_{i}\geq w_{i+1}$ for every $i\in\\{0,\dots,l-1\\}$. If $l\delta\geq w_{l}$,
then following statements hold.

  1. (i)

$\gamma_{w}(G)\leq\gamma_{w}^{s}(G).$

  2. (ii)

If $k\in\mathbb{Z}^{+}$, then
$\gamma_{(k+1,k=w_{1},\dots,w_{l})}(G)\leq\gamma_{(k,k=w_{1},\dots,w_{l})}^{s}(G)$.

######  Theorem 2.2.

[5] Let $G$ be a graph of minimum degree $\delta$, and let
$w=(w_{0},\dots,w_{l}),w^{\prime}=(w_{0}^{\prime},\dots,w^{\prime}_{l})\in\mathbb{Z}^{+}\times\mathbb{N}^{l}$
such that $l\delta\geq w_{l}$, $w_{i}\geq w_{i+1}$ and $w_{i}^{\prime}\geq
w_{i+1}^{\prime}$ for every $i\in\\{0,\dots,l-1\\}$. If $w_{i}\geq
w_{i-1}^{\prime}-1$ for every $i\in\\{1,\dots,l\\}$, and
$\max\\{w_{j}-1,0\\}\geq w_{j}^{\prime}\,$ for every $j\in\\{0,\dots,l\\}$,
then

$\gamma_{w^{\prime}}^{s}(G)\leq\gamma_{w}(G).$

The following result is a particular case of Theorem 2.2.

######  Corollary 2.3.

[5] Let $G$ be a graph of minimum degree $\delta$, and let
$w=(w_{0},\dots,w_{l})\in\mathbb{Z}^{+}\times\mathbb{N}^{l}$ and
$\textbf{1}=(1,\dots,1)$. If $0\leq w_{j-1}-w_{j}\leq 2$ for every
$j\in\\{1,\dots,i\\}$, where $1\leq i\leq l$ and $l\delta\geq w_{l}+1$, then

$\gamma_{(w_{0},\dots,w_{i},0,\dots,0)}^{s}(G)\leq\gamma_{(w_{0}+1,\dots,w_{i}+1,0,\dots,0)}(G)\leq\gamma_{w+\textbf{1}}(G).$

######  Proposition 2.4.

[5] Let $G$ be a graph of order $n$. Let
$w=(w_{0},\dots,w_{l})\in\mathbb{Z}^{+}\times\mathbb{N}^{l}$ such that
$w_{0}\geq\cdots\geq w_{l}$. If $G^{\prime}$ is a spanning subgraph of $G$
with minimum degree $\delta^{\prime}\geq\frac{w_{l}}{l}$, then

$\gamma_{w}^{s}(G)\leq\gamma_{w}^{s}(G^{\prime}).$

##  3 The case of lexicographic product graphs

The _lexicographic product_ of two graphs $G$ and $H$ is the graph $G\circ H$
whose vertex set is $V(G\circ H)=V(G)\times V(H)$ and $(u,v)(x,y)\in E(G\circ
H)$ if and only if $ux\in E(G)$ or $u=x$ and $vy\in E(H)$.

Notice that for any $u\in V(G)$ the subgraph of $G\circ H$ induced by
$\\{u\\}\times V(H)$ is isomorphic to $H$. For simplicity, we will denote this
subgraph by $H_{u}$. Moreover, the neighbourhood of $(x,y)\in V(G)\times V(H)$
will be denoted by $N(x,y)$ instead of $N((x,y))$. Analogously, for any
function $f$ on $G\circ H$, the image of $(x,y)$ will be denoted by $f(x,y)$
instead of $f((x,y))$.

The next subsections are devoted to show how the secure (total) domination
number and the (total) weak Roman domination number of lexicographic product
graphs $G\circ H$ are related to $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$, for
certain vectors $w$ of three components.

###  3.1 Secure domination

######  Lemma 3.1.

For any graph $G$ with no isolated vertex and any nontrivial graph $H$, there
exists a $\gamma_{(1,0)}^{s}(G\circ H)$-function $f$ such that
$f(V(H_{u}))\leq 2$ for every $u\in V(G)$.

###### Proof.

Given a secure $(1,0)$-dominating function $f$ on $G\circ H$, we define

$R_{f}=\\{x\in V(G):\,f(V(H_{x}))\geq 3\\}.$

Let $f$ be a $\gamma_{(1,0)}^{s}(G\circ H)$-function such that $|R_{f}|$ is
minimum among all $\gamma_{(1,0)}^{s}(G\circ H)$-functions.

Suppose that $|R_{f}|\geq 1$. Let $u\in R_{f}$, $u^{\prime}\in N(u)$ and
$v_{1},v_{2}\in V(H)$ such that $f(u,v_{1})=f(u,v_{2})=1$. Now, let
$f^{\prime}:V(G)\times V(H)\longrightarrow\\{0,1\\}$ be a function defined as
follows.

  * $\bullet$

$f^{\prime}(u,v_{1})=f^{\prime}(u,v_{2})=1$ and $f^{\prime}(u,y)=0$ for every
$y\in V(H)\setminus\\{v_{1},v_{2}\\}$;

  * $\bullet$

$f^{\prime}(V(H_{u^{\prime}}))=\min\\{2,f(V(H_{u^{\prime}}))+f(V(H_{u}))-2\\}$;

  * $\bullet$

$f^{\prime}(x,y)=f(x,y)$ for every $x\in V(G)\setminus\\{u,u^{\prime}\\}$ and
$y\in V(H)$.

It is not difficult to check that $f^{\prime}$ is a secure $(1,0)$-dominating
function on $G\circ H$ with $\omega(f^{\prime})\leq\omega(f)$ and
$|R_{f^{\prime}}|<|R_{f}|$, which is a contradiction. Therefore,
$R_{f}=\varnothing$, and the result follows. ∎

We shall need the following two results.

######  Theorem 3.2.

[18] For any graph $G$ with no isolated vertex and any nontrivial graph $H$
with $\gamma_{(1,0)}^{s}(H)\leq 2$ or $\gamma_{(1,0,0)}^{s}(H)\geq 3$,

$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(1,0,0)}^{s}(G\circ H).$

######  Proposition 3.3.

[22] For any graph $G$ and any integer $n\geq 1$,

$\gamma_{(1,0,0)}^{s}(G\circ K_{n})=\gamma_{(1,0,0)}^{s}(G)$

The following result shows how the secure domination number of $G\circ H$ is
related to $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$ for certain vectors $w$ of
three components. The decision on whether the components of $w$ take specific
values will depend on the value of $\gamma_{(1,0)}^{s}(H)$ and $\gamma(H)$.

######  Theorem 3.4.

For a graph $G$ with no isolated vertex and a nontrivial graph $H$, the
following statements hold.

  1. (i)

If $\gamma_{(1,0)}^{s}(H)=1$, i.e., $H$ is a complete graph, then
$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(1,0,0)}^{s}(G)$.

  2. (ii)

If $\gamma_{(1,0)}^{s}(H)=2$ and $\gamma(H)=1$, then
$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(2,1,0)}(G)$.

  3. (iii)

If $\gamma_{(1,0)}^{s}(H)\geq 3$ and $\gamma(H)=1$, then
$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(2,1,1)}(G)$.

  4. (iv)

If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$, then
$\gamma_{(1,1,0)}^{s}(G)\leq\gamma_{(1,0)}^{s}(G\circ
H)\leq\gamma_{(2,2,0)}(G)$.

  5. (v)

If $\gamma_{(1,0)}^{s}(H)>\gamma(H)=2$, then $\gamma_{(1,0)}^{s}(G\circ
H)=\gamma_{(2,2,1)}(G)$.

  6. (vi)

If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$, then
$\gamma_{(2,2,1)}(G)\leq\gamma_{(1,0)}^{s}(G\circ H)\leq\gamma_{(2,2,2)}(G)$.

  7. (vii)

If $\gamma_{(1,0)}^{s}(H)\geq 4$ and $\gamma(H)\geq 3$, then
$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(2,2,2)}(G)$.

###### Proof.

Let $f(V_{0},V_{1})$ be a $\gamma_{(1,0)}^{s}(G\circ H)$-function which
satisfies Lemma 3.1. Let $f^{\prime}(X_{0},X_{1},X_{2})$ be the function
defined on $G$ by $X_{1}=\\{x\in V(G):f(V(H_{x}))=1\\}$ and $X_{2}=\\{x\in
V(G):f(V(H_{x}))=2\\}$. Notice that $\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})$. With this notation in mind, we differentiate
the following cases.

Case $1.$ $\gamma_{(1,0)}^{s}(H)=1$. In this case, $H$ is a complete graph.
Hence, by Theorem 3.2 and Proposition 3.3 we deduce that
$\gamma_{(1,0)}^{s}(G\circ H)=\gamma_{(1,0,0)}^{s}(G\circ
H)=\gamma_{(1,0,0)}^{s}(G)$.

Case $2.$ $\gamma_{(1,0)}^{s}(H)=2$ and $\gamma(H)=1$. In this case, if $x\in
X_{0}$, then $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 2$. Now,
since $\gamma_{(1,0)}^{s}(H)=2$, if $x\in X_{1}$, then
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$. Therefore,
$f^{\prime}$ is a $(2,1,0)$-dominating function on $G$, which implies that
$\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,1,0)}(G)$.

On the other side, for any $\gamma_{(2,1,0)}(G)$-function
$g(W_{0},W_{1},W_{2})$ and any universal vertex $v$ of $H$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime},W_{2}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v\\}$ and $W_{2}^{\prime}=W_{2}\times\\{v\\}$,
is a $(2,1,0)$-dominating function on $G\circ H$. Hence,
$\gamma_{(2,1,0)}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,1,0)}(G)$. Therefore, by Theorem
3.2 and Corollary 2.3 we conclude that $\gamma_{(1,0)}^{s}(G\circ
H)=\gamma_{(1,0,0)}^{s}(G\circ H)\leq\gamma_{(2,0,0)}(G\circ
H)\leq\gamma_{(2,1,0)}(G\circ H)\leq\gamma_{(2,1,0)}(G)$.

Case $3.$ $\gamma_{(1,0)}^{s}(H)\geq 3$ and $\gamma(H)=1$. As above, if $x\in
X_{0}$, then $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 2$ and,
since $\gamma_{(1,0)}^{s}(H)\geq 3$, if $x\in X_{1}\cup X_{2}$, then
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$. Hence, $f^{\prime}$
is a $(2,1,1)$-dominating function on $G$. Therefore,
$\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,1,1)}(G)$.

On the other side, for any $\gamma_{(2,1,1)}(G)$-function
$g(W_{0},W_{1},W_{2})$, any universal vertex $v$ of $H$ and any $v^{\prime}\in
V(H)\setminus\\{v\\}$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v\\}\cup W_{2}\times\\{v,v^{\prime}\\}$, is a
$(1,0)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}$ with $x\in W_{1}\cup W_{2}$, we can see that
$g^{\prime}_{(x,v)\rightarrow(x,y)}$ is a $(1,0)$-dominating function on
$G\circ H$, while for any $(x,y)\in W_{0}\times V(H)$ there exists
$x^{\prime}\in W_{2}\cap N(x)$ or $x^{\prime},x^{\prime\prime}\in W_{1}\cap
N(x)$, and so $g^{\prime}_{(x^{\prime},v)\rightarrow(x,y)}$ is a
$(1,0)$-dominating function on $G\circ H$. Therefore, $g^{\prime}$ is a secure
$(1,0)$-dominating function on $G\circ H$ and, as a consequence,
$\gamma_{(1,0)}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,1,1)}(G)$.

Case $4.$ $\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$. If $x\in X_{0}\cup X_{1}$, then
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$, which implies that
$f^{\prime}$ is a $(1,1,0)$-dominating function on $G$. Now, for any $(x,y)\in
X_{0}\times V(H)$, there exists $(x^{\prime},y^{\prime})\in M_{f}(x,y)$ with
$x^{\prime}\in N(x)\cap(X_{1}\cup X_{2})$. Hence, for any $u\in
X_{0}\cup\\{x^{\prime}\\}$ we have that $f^{\prime}_{x^{\prime}\rightarrow
x}(N(u))=f_{(x^{\prime},y^{\prime})\rightarrow(x,y)}(N(V(H_{u}))\setminus
V(H_{u}))\geq 1$. Now, if $u\in X_{1}$, then as $\gamma_{(1,0)}^{s}(H_{u})=2$
we have that $f^{\prime}_{x^{\prime}\rightarrow
x}(N(u))=f_{(x^{\prime},y^{\prime})\rightarrow(x,y)}(N(V(H_{u}))\setminus
V(H_{u}))\geq 1$. Hence, $f^{\prime}$ is a secure $(1,1,0)$-dominating
function on $G$. Therefore, $\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(1,1,0)}^{s}(G)$.

On the other side, for any $\gamma_{(2,2,0)}(G)$-function
$g(W_{0},W_{1},W_{2})$ and any $\gamma_{(1,0)}^{s}(H)$-function
$h(Y_{0},Y_{1})$ with $Y_{1}=\\{v_{1},v_{2}\\}$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v_{1}\\}\cup W_{2}\times Y_{1}$, is a
$(1,0)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}\setminus W_{2}\times V(H)$ and $x^{\prime}\in
N(x)\cap(W_{1}\cup W_{2})$, we can see that
$g^{\prime}_{(x^{\prime},v_{1})\rightarrow(x,y)}$ is a $(1,0)$-dominating
function on $G\circ H$. Furthermore, for every $(x,y)\in W_{0}^{\prime}\cap
W_{2}\times V(H)$ there exists $v_{i}\in\\{Y_{1}\\}\cap M_{h}(y)$ such that
$g^{\prime}_{(x,v_{i})\rightarrow(x,y)}$ is a $(1,0)$-dominating function on
$G\circ H$. Therefore, $\gamma_{(1,0)}^{s}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,2,0)}(G)$.

Case $5.$ $\gamma_{(1,0)}^{s}(H)>\gamma(H)=2$. Since
$\gamma_{(1,0)}^{s}(H)\geq 3$, if $x\in X_{0}\cup X_{1}$, then
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 2$, while if $x\in
X_{2}$, then $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$. Hence,
$f^{\prime}$ is a $(2,2,1)$-dominating function on $G$. Therefore,
$\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,2,1)}(G)$.

In order to prove that $\gamma_{(1,0)}^{s}(G\circ H)\leq\gamma_{(2,2,1)}(G)$,
let $g(W_{0},W_{1},W_{2})$ be a $\gamma_{(2,2,1)}(G)$-function. If
$\gamma_{(1,0)}^{s}(H)>\gamma(H)=2$, then for any dominating set
$S=\\{v_{1},v_{2}\\}$ of $H$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v_{1}\\}\cup W_{2}\times S$, is a
$(2,1)$-dominating function on $G\circ H$. Hence, $\gamma_{(2,1)}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,2,1)}(G)$, and so Corollary 2.3
leads to $\gamma_{(1,0)}^{s}(G\circ H)\leq\gamma_{(2,1)}(G\circ
H)\leq\gamma_{(2,2,1)}(G).$

Case $6.$ $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$. As in Case 5, we deduce that
$\gamma_{(1,0)}^{s}(G\circ H)\geq\gamma_{(2,2,1)}(G)$.

In order to prove that $\gamma_{(1,0)}^{s}(G\circ H)\leq\gamma_{(2,2,2)}(G)$,
let $g(W_{0},W_{1},W_{2})$ be a $\gamma_{(2,2,2)}(G)$-function and
$S=\\{v_{1},v_{2}\\}\subseteq V(H)$. The function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v_{1}\\}\cup W_{2}\times S$, is a
$(2,2)$-dominating function on $G\circ H$. Hence, $\gamma_{(2,2)}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,2,2)}(G)$. Therefore, Corollary
2.3 leads to $\gamma_{(1,0)}^{s}(G\circ H)\leq\gamma_{(1,1)}^{s}(G\circ
H)\leq\gamma_{(2,2)}(G\circ H)\leq\gamma_{(2,2,2)}(G).$

Case $7.$ $\gamma_{(1,0)}^{s}(H)\geq 4$ and $\gamma(H)\geq 3$. In this case,
it is easy to check that $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus
V(H_{x}))\geq 2$ for every $x\in V(G)$. Therefore, $f^{\prime}$ is a
$(2,2,2)$-dominating function on $G$, which implies that
$\gamma_{(1,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,2,2)}(G)$.

Finally, as in Case 6, we can deduce that $\gamma_{(2,2)}(G\circ
H)\leq\gamma_{(2,2,2)}(G)$, and so $\gamma_{(1,0)}^{s}(G\circ
H)\leq\gamma_{(1,1)}^{s}(G\circ H)\leq\gamma_{(2,2)}(G\circ
H)\leq\gamma_{(2,2,2)}(G).$ ∎

$1$$1$$1$$1$$1$$1$ Figure 1: A graph $G$, where the labels asigned to the
vertices correspond to the positive weights assigned by a
$\gamma_{(1,1,0)}^{s}(G)$-function.

$2$$2$$2$$G_{1}$

$1$$2$$1$$2$$G_{2}$

$2$$2$$2$$1$$2$$1$$2$$G_{3}$

Figure 2: For any $i\in\\{1,2,3\\}$, the labels asigned to the vertices of
$G_{i}$ correspond to the positive weights assigned by a
$\gamma_{(1,0)}^{s}(G_{i}\circ H)$-function to the different copies of $H$ in
$G_{i}\circ H$, where $H$ is any graph with
$\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$.

In order to show the behaviour of $\gamma_{(1,0)}^{s}(G\circ H)$ when
$\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$, we consider the following examples. For
the graph $G$ shown in Figure 1, $\gamma_{(1,0)}^{s}(G\circ
H)=\gamma_{(1,1,0)}^{s}(G)=6<8=\gamma_{(2,2,0)}(G)$, while
$\gamma_{(1,1,0)}^{s}(G\cup C_{4})=9<10=\gamma_{(1,0)}^{s}((G\cup C_{4})\circ
H)<12=\gamma_{(2,2,0)}(G\cup C_{4})$ and $\gamma_{(1,0)}^{s}(C_{4}\circ
H)=\gamma_{(2,2,0)}(C_{4})=4>3=\gamma_{(1,1,0)}^{s}(C_{4})$.

Analogously, for the case $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$, we consider the
graphs $G_{1}$, $G_{2}$ and $G_{3}$ illustrated in Figure 2. The weights shown
in Figure 2 correspond to the weights assigned by a
$\gamma_{(1,0)}^{s}(G_{i}\circ H)$-function to the different copies of $H$ in
$G_{i}\circ H$. In particular, $\gamma_{(1,0)}^{s}(G_{1}\circ
H)=\gamma_{(2,2,2)}(G_{1})=6$, $\gamma_{(1,0)}^{s}(G_{2}\circ
H)=\gamma_{(2,2,1)}(G_{2})=6$ and
$\gamma_{(2,2,1)}(G_{3})=11<12=\gamma_{(1,0)}^{s}(G_{3}\circ
H)<14=\gamma_{(2,2,2)}(G_{3}).$

We now discuss some particular cases of Theorem 3.4.

######  Corollary 3.5.

The following statements hold for any integers $n,r\geq 4$ and a nontrivial
graph $H$.

  * •

$\gamma_{(1,0)}^{s}(K_{n}\circ H)=\left\\{\begin{array}[]{ll}1&\text{if
}H\text{ is a complete graph},\\\\[5.0pt] 2&\text{if
}\,\gamma_{(1,0)}^{s}(H)>\gamma(H)=1\,\text{ or
}\,\gamma_{(1,0)}^{s}(H)=\gamma(H)=2,\\\\[5.0pt]
3&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,0)}^{s}(K_{1,n-1}\circ H)=\left\\{\begin{array}[]{ll}2&\text{if
}\,\gamma_{(1,0)}^{s}(H)\leq 2,\\\\[5.0pt] 4&\text{if
}\,\gamma_{(1,0)}^{s}(H)\geq 4\,\text{ and }\,\gamma(H)\geq 3,\\\\[5.0pt]
3&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,0)}^{s}(K_{2,n}\circ H)=\left\\{\begin{array}[]{ll}2&\text{if
}H\text{ is a complete graph},\\\\[5.0pt] 4&\text{if
}\,\gamma_{(1,0)}^{s}(H)\geq 2\,\text{ and }\,\gamma(H)\geq 2,\\\\[5.0pt]
3&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,0)}^{s}(K_{3,n}\circ H)=\left\\{\begin{array}[]{ll}3&\text{if
}H\text{ is a complete graph},\\\\[5.0pt]
4&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,0)}^{s}(K_{n,r}\circ H)=4$.

###### Proof.

It is not difficult to see that if $\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$, then
$\gamma_{(1,0)}^{s}(K_{2,n}\circ H)=\gamma_{(2,2,0)}(K_{2,n})=4$. For the
remaining cases, the result follows from Theorem 3.4 by considering the
following facts.

  * •

$\gamma_{(1,0,0)}^{s}(K_{n})=1$,
$\gamma_{(2,1,0)}(K_{n})=\gamma_{(2,1,1)}(K_{n})=\gamma_{(1,1,0)}^{s}(K_{n})=\gamma_{(2,2,0)}(K_{n})=2$
and $\gamma_{(2,2,1)}(K_{n})=\gamma_{(2,2,2)}(K_{n})=3$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{1,n-1})=\gamma_{(2,1,0)}(K_{1,n-1})=\gamma_{(1,1,0)}^{s}(K_{1,n-1})=\gamma_{(2,2,0)}(K_{1,n-1})=2$,
$\gamma_{(2,1,1)}(K_{1,n-1})=\gamma_{(2,2,1)}(K_{1,n-1})=3$ and
$\gamma_{(2,2,2)}(K_{1,n-1})=4$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{2,n})=2$,
$\gamma_{(2,1,0)}(K_{2,n})=\gamma_{(2,1,1)}(K_{2,n})=3$ and
$\gamma_{(2,2,1)}(K_{2,n})=\gamma_{(2,2,2)}(K_{2,n})=4$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{3,n})=3$ and
$\gamma_{(2,1,0)}(K_{3,n})=\gamma_{(2,1,1)}(K_{3,n})=\gamma_{(1,1,0)}^{s}(K_{3,n})=\gamma_{(2,2,0)}(K_{3,n})=\gamma_{(2,2,1)}(K_{3,n})=\gamma_{(2,2,2)}(K_{3,n})=4$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{n,r})=\gamma_{(2,1,0)}(K_{n,r})=\gamma_{(1,1,0)}^{s}(K_{n,r})=\gamma_{(2,2,0)}(K_{n,r})=\gamma_{(2,1,1)}(K_{n,r})=\gamma_{(2,2,1)}(K_{n,r})=\gamma_{(2,2,2)}(K_{n,r})=4$.

∎

Next we consider the particular case when $G$ is a path. As we will see, this
case has been partially studied in previous works.

######  Theorem 3.6.

For any integer $n\geq 6$ and any nontrivial graph $H$, the following
statements hold.

  1. (i)

[17] If $\gamma_{(1,0)}^{s}(H)=1$, i.e., $H$ is a complete graph, then
$\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\gamma_{(1,0,0)}^{s}(P_{n})=\left\lceil\frac{3n}{7}\right\rceil$.

  2. (ii)

[18] If $\gamma_{(1,0)}^{s}(H)=2$ and $\gamma(H)=1$, then
$\gamma_{(1,0)}^{s}(P_{n}\circ H)=2\left\lceil\frac{n}{3}\right\rceil$.

  3. (iii)

[18] If $\gamma_{(1,0)}^{s}(H)\geq 3$ and $\gamma(H)=1$, then
$\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\left\\{\begin{array}[]{ll}\frac{2n}{3}+1&\text{if }\,n\equiv
0\pmod{3},\\\\[5.0pt]
2\lceil\frac{n}{3}\rceil&\mbox{otherwise.}\end{array}\right.$

  4. (iv)

[18] If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$, then
$\gamma_{(1,0)}^{s}(P_{n}\circ H)=2\left\lfloor\frac{n+2}{3}\right\rfloor$.

  5. (v)

If $\gamma_{(1,0)}^{s}(H)>\gamma(H)=2$, then

$\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\gamma_{(2,2,1)}(P_{n})=\left\\{\begin{array}[]{ll}n-\lfloor\frac{n}{7}\rfloor+1&\text{if
}\,n\equiv 1,2\pmod{7},\\\\[5.0pt]
n-\lfloor\frac{n}{7}\rfloor&\mbox{otherwise.}\end{array}\right.$

  6. (vi)

If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$, then

$\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\left\\{\begin{array}[]{ll}n-\left\lfloor\frac{n}{11}\right\rfloor+1&\text{if
}\,n\equiv 1,2,5\pmod{11},\\\\[5.0pt]
n-\left\lfloor\frac{n}{11}\right\rfloor&\mbox{otherwise.}\end{array}\right.$

  7. (vii)

If $\gamma_{(1,0)}^{s}(H)\geq 4$ and $\gamma(H)\geq 3$, then

$\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\gamma_{(2,2,2)}(P_{n})=\left\\{\begin{array}[]{ll}n&if\,n\equiv
0\pmod{4},\\\\[5.0pt] n+1&if\,n\equiv 1,3\pmod{4},\\\\[5.0pt] n+2&if\,n\equiv
2\pmod{4}.\end{array}\right.$

###### Proof.

The proofs of (v) and (vii) are derived by combining Theorem 3.4 with the
values of $\gamma_{(2,2,1)}(P_{n})$ and $\gamma_{(2,2,2)}(P_{n})$ obtained in
[4]. It remains to prove (vi).

Assume that $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$ and let $f(V_{0},V_{1})$ be a
$\gamma_{(1,0)}^{s}(P_{n}\circ H)$-function which satisfies Lemma 3.1. Let
$f^{\prime}(X_{0},X_{1},X_{2})$ be the function defined on $P_{n}$ by
$X_{1}=\\{x\in V(P_{n}):f(V(H_{x}))=1\\}$ and $X_{2}=\\{x\in
V(P_{n}):f(V(H_{x}))=2\\}$. Notice that $\gamma_{(1,0)}^{s}(P_{n}\circ
H)=\omega(f)=\omega(f^{\prime})$. Let $n=11q+r$ with $r\in\\{0,\dots,10\\}$.
With this notation in mind, we proceed by induction on $q$.

It is not difficult to check that the result follows for $q=0$ and
$r\in\\{6,\dots,10\\}$. For these cases, possible sequences of weights
assigned by $f^{\prime}$ to consecutive vertices of $P_{r}$ are $021120$,
$1200220$, $02200220$, $021012120$ and $0210121012$, respectively. The result
also follows for $q=1$ and $r=0$, i.e., $n=11$. In this case, the only
possible sequence of weights assigned by $f^{\prime}$ to consecutive vertices
of $P_{11}$ is $02101210120$. The certainty that the result holds for $q=1$
and $r\in\\{3,4,6,\dots,10\\}$ comes from a computer search. For these cases,
since $P_{11+r}$ can be obtained by connecting a leaf of $P_{11}$ with a leaf
$P_{r}$, possible sequences of weights assigned by $f^{\prime}$ to consecutive
vertices of $P_{11+r}$ are obtained by concatenating the sequences of weights
associated to $P_{11}$ and $P_{r}$, where the sequences for $r=3$ and $r=4$
are $120$, $0220$, respectively. In summary, for any
$r\in\\{0,3,4,6,\dots,10\\}$, we have that $\gamma_{(1,0)}^{s}(P_{11+r}\circ
H)=10+r$. Analogously, among the possible sequences of weights assigned by
$f^{\prime}$ to consecutive vertices of $P_{12}$, $P_{13}$ and $P_{16}$, a
computer search gives, for instance, $022101200220$, $0210210220120$ and
$0211200220021120$, respectively. Thus, $\gamma_{(1,0)}^{s}(P_{12}\circ
H)=12$, $\gamma_{(1,0)}^{s}(P_{13}\circ H)=13$ and
$\gamma_{(1,0)}^{s}(P_{16}\circ H)=16$, which completes the base case.

Assume that $q\geq 2$ and the statement holds for any $q^{\prime}$ such that
$1\leq q^{\prime}\leq q$. We differentiate two cases.

Case $1.$ $r\not\in\\{1,2,5\\}$. In this case,

$\gamma_{(1,0)}^{s}(P_{11q+r}\circ H)=10q+r.$

Let $n=11(q+1)+r$, $k_{1}=\gamma_{(1,0)}^{s}(P_{n}\circ H)$,
$P_{n}=x_{1}x_{2}\dots x_{n}$ and
$k_{2}=f^{\prime}(\\{x_{1},\dots,x_{11}\\})$. Suppose that $k_{1}<10(q+1)+r$.
Since $02101210120$ is the only possible sequence of weights assigned by
$f^{\prime}$ to consecutive vertices of $P_{11}$, we have that $k_{2}\geq 10$,
and

$(k_{2}-10)+f^{\prime}(\\{x_{12},\dots,x_{n}\\})=k_{1}-10<10q+r.$

Since the function $g$, defined as
$g(V(H_{x_{13}}))=f(V(H_{x_{13}}))+k_{2}-10$ and
$g(V(H_{x_{i}}))=f(V(H_{x_{i}}))$ for every
$i\in\\{12\\}\cup\\{14,\dots,n\\}$, is a secure $(1,0)$-dominating function on
the subgraph of $P_{n}\circ H$ induced by $\\{x_{12},\dots,x_{n}\\}\times
V(H)$, we can conclude that $\gamma_{(1,0)}^{s}(P_{11q+r}\circ
H)\leq\omega(g)=k_{1}-10<10q+r$, which contradicts the hypothesis. Thus,
$\gamma_{(1,0)}^{s}(P_{n}\circ H)\geq 10(q+1)+r$. To conclude the proof, we
only need to observe that $P_{n}$ is obtained by connecting a leaf of
$P_{11q+r}$ with a leaf of $P_{11}$, and so, by hypothesis,
$\gamma_{(1,0)}^{s}(P_{n}\circ H)\leq\gamma_{(1,0)}^{s}(P_{11q+r}\circ
H)+\gamma_{(1,0)}^{s}(P_{11}\circ H)=10(q+1)+r$. Therefore, the proof of this
case is complete.

Case $2.$ $r\in\\{1,2,5\\}$. This case is completely analogous to Case 1. The
only difference is the induction hypothesis, which states that
$\gamma_{(1,0)}^{s}(P_{11q+r}\circ H)=10q+r+1.$ Hence, we take $n=11(q+1)+r$
and following the procedure described above, we deduce that
$\gamma_{(1,0)}^{s}(P_{n}\circ H)=10(q+1)+r+1$, which completes the proof. ∎

The next result concerns the case when $G$ is a cycle.

######  Theorem 3.7.

For any integer $n\geq 6$ and a nontrivial graph $H$, the following statements
hold.

  1. (i)

[17] If $\gamma_{(1,0)}^{s}(H)=1$, i.e., $H$ is a complete graph, then
$\gamma_{(1,0)}^{s}(C_{n}\circ
H)=\gamma_{(1,0,0)}^{s}(C_{n})=\left\lceil\frac{3n}{7}\right\rceil$.

  2. (ii)

[18] If $\gamma_{(1,0)}^{s}(H)\geq 2$ and $\gamma(H)=1$, then
$\gamma_{(1,0)}^{s}(C_{n}\circ H)=\left\lceil\frac{2n}{3}\right\rceil$.

  3. (iii)

[18] If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=2$, then
$\gamma_{(1,0)}^{s}(C_{n}\circ H)=2\left\lfloor\frac{n+2}{3}\right\rfloor$.

  4. (iv)

If $\gamma_{(1,0)}^{s}(H)>\gamma(H)=2$, then

$\gamma_{(1,0)}^{s}(C_{n}\circ
H)=\gamma_{(2,2,1)}(C_{n})=\left\\{\begin{array}[]{ll}n-\lfloor\frac{n}{7}\rfloor+1&\text{if
}\,n\equiv 1,2\pmod{7},\\\\[5.0pt]
n-\lfloor\frac{n}{7}\rfloor&\mbox{otherwise.}\end{array}\right.$

  5. (v)

If $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$, then

$\gamma_{(1,0)}^{s}(C_{n}\circ
H)=\left\\{\begin{array}[]{ll}n-\left\lfloor\frac{n}{11}\right\rfloor+1&\text{if
}\,n\equiv 1,2,5\pmod{11},\\\\[5.0pt]
n-\left\lfloor\frac{n}{11}\right\rfloor&\mbox{otherwise.}\end{array}\right.$

  6. (vi)

If $\gamma_{(1,0)}^{s}(H)\geq 4$ and $\gamma(H)\geq 3$, then
$\gamma_{(1,0)}^{s}(C_{n}\circ H)=\gamma_{(2,2,2)}(C_{n})=n.$

###### Proof.

The proofs of (iv) and (vi) are derived by combining Theorem 3.4 with the
values of $\gamma_{(2,2,1)}(C_{n})$ and $\gamma_{(2,2,2)}(C_{n})$ obtained in
[4]. It remains to prove (v).

Assume that $\gamma_{(1,0)}^{s}(H)=\gamma(H)=3$ and let $f(V_{0},V_{1})$ be a
$\gamma_{(1,0)}^{s}(C_{n}\circ H)$-function which satisfies Lemma 3.1. Let
$f^{\prime}(X_{0},X_{1},X_{2})$ be the function defined on $C_{n}$ by
$X_{1}=\\{x\in V(C_{n}):f(V(H_{x}))=1\\}$ and $X_{2}=\\{x\in
V(C_{n}):f(V(H_{x}))=2\\}$. Notice that $\gamma_{(1,0)}^{s}(C_{n}\circ
H)=\omega(f)=\omega(f^{\prime})$.

Let $V(C_{n})=\\{x_{0},\dots,x_{n-1}\\}$, where consecutive vertices are
adjacent and the addition of subscripts is taken modulo $n$. If there exists
$x_{i}\in V(C_{n})$ such that $f(x_{i})=f(x_{i+1})=0$, then
$\gamma_{(1,0)}^{s}(C_{n}\circ H)=\gamma_{(1,0)}^{s}(P_{n}\circ H)$ and we
derive the result by Theorem 3.6. From now on, we assume that for any
$x_{i}\in X_{0}$ we have that $f^{\prime}(x_{i-1})>0$ and
$f^{\prime}(x_{i+1})>0$, which implies that
$f^{\prime}(x_{i-2})+f(x_{i-1})\geq 3$ and $f^{\prime}(x_{i+1})+f(x_{i+2})\geq
3$. Now, for any $x_{i}\in X_{0}$ we define
$S_{i}=\\{x_{i},x_{i+1},x_{i+2}\\}$ and
$S=V(C_{n})\setminus\left(\cup_{x_{i}\in X_{0}}S_{i}\right)$. Notice that
$f(S_{i})\geq 3=|S_{i}|$ for every $x_{i}\in X_{0}$ and $f(x_{j})>0$ for every
$x_{j}\in S$. Hence, by Proposition 2.4,

$\gamma_{(1,0)}^{s}(P_{n}\circ H)\geq\gamma_{(1,0)}^{s}(C_{n}\circ
H)=\omega(f^{\prime})=\sum_{x_{i}\in X_{0}}f^{\prime}(S_{i})+f^{\prime}(S)\geq
n.$

This implies that $n\in\\{6,\dots,10,12,13,15\\}$ and
$\gamma_{(1,0)}^{s}(C_{n}\circ H)=n$. Therefore, the result follows. ∎

As a direct consequence of Theorems 3.4, 3.6 and 3.7 we derive the following
result.

######  Proposition 3.8.

The following statements hold for any integer $n\geq 4$.

  * •

$\gamma_{(2,1,1)}(P_{n})=\left\\{\begin{array}[]{ll}\frac{2n}{3}+1&\text{if
}\,n\equiv 0\pmod{3},\\\\[5.0pt]
2\lceil\frac{n}{3}\rceil&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(2,1,1)}(C_{n})=\lceil\frac{2n}{3}\rceil.$

###  3.2 Weak Roman domination

This subsection is devoted to study the weak Roman domination of lexicographic
product graphs. To this end, we need the following tools.

######  Remark 3.9.

[22] Given a noncomplete graph $G$, the following statements are equivalent.

  * •

$\gamma_{(1,0,0)}^{s}(G)=2.$

  * •

$\gamma(G)=1$ or $\gamma_{(1,0)}^{s}(G)=2.$

######  Lemma 3.10.

For any graph $G$ with no isolated vertex and any nontrivial graph $H$ with
$\gamma(H)=1$, there exists a $\gamma_{(1,0,0)}^{s}(G\circ H)$-function $f$
such that $f(V(H_{u}))\leq 2$, for every $u\in V(G)$.

###### Proof.

Given a secure $(1,0,0)$-dominating function $f$ on $G\circ H$, we define
$R_{f}=\\{x\in V(G):\,f(V(H_{x}))\geq 3\\}$. Let $f(V_{0},V_{1},V_{2})$ be a
$\gamma_{(1,0,0)}^{s}(G\circ H)$-function such that $|R_{f}|$ is minimum among
all $\gamma_{(1,0,0)}^{s}(G\circ H)$-functions.

Suppose that $|R_{f}|\geq 1$. Let $u\in R_{f}$, $u^{\prime}\in N(u)$ and $v$
be a universal vertex of $H$. Notice that the function $f^{\prime}$ on $G\circ
H$, defined by $f^{\prime}(u,v)=f^{\prime}(V(H_{u}))=2$,
$f^{\prime}(V(H_{u^{\prime}}))=\min\\{2,f(V(H_{u^{\prime}}))+1\\}$, and
$f^{\prime}(x,y)=f(x,y)$ for every $x\in V(G)\setminus\\{u,u^{\prime}\\}$ and
$y\in V(H)$, is a secure $(1,0,0)$-dominating function on $G\circ H$ with
$\omega(f^{\prime})\leq\omega(f)$ and $|R_{f^{\prime}}|<|R_{f}|$, which is a
contradiction. Therefore, $R_{f}=\varnothing$, and so the result follows. ∎

Our goal is to show how the weak Roman domination number of $G\circ H$ is
related to $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$ for certain vectors $w$ of
three components. By Theorems 3.2 and 3.4, the outstanding case is when
$\gamma_{(1,0)}^{s}(H)\geq 3$ and $\gamma_{(1,0,0)}^{s}(H)=2$, which is
equivalent to $\gamma(H)=1$ and $\gamma_{(1,0)}^{s}(H)\geq 3$, by Remark 3.9.
In fact, we will present the result with the only assumption on $H$ of being a
noncomplete graph with $\gamma(H)=1$.

######  Theorem 3.11.

For any graph $G$ with no isolated vertex and any noncomplete graph $H$ with
$\gamma(H)=1$,

$\gamma_{(1,0,0)}^{s}(G\circ H)=\gamma_{(2,1,0)}(G).$

###### Proof.

Let $H$ be a noncomplete graph with $\gamma(H)=1$ and $f(V_{0},V_{1},V_{2})$ a
$\gamma_{(1,0,0)}^{s}(G\circ H)$-function which satisfies Lemma 3.10. Let
$f^{\prime}(X_{0},X_{1},X_{2})$ be the function defined on $G$ by
$X_{1}=\\{x\in V(G):f(V(H_{x}))=1\\}$ and $X_{2}=\\{x\in
V(G):f(V(H_{x}))=2\\}$. Since $\gamma_{(1,0,0)}^{s}(H)=2$, if $x\in X_{0}$,
then $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 2$ and if $x\in
X_{1}$, then $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$. Hence,
$f^{\prime}$ is a $(2,1,0)$-dominating function on $G$, which implies that
$\gamma_{(1,0,0)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,1,0)}(G)$.

Now, we show that $\gamma_{(1,0,0)}^{s}(G\circ H)\leq\gamma_{(2,1,0)}(G)$. Let
$v$ be a universal vertex of $H$. For any $\gamma_{(2,1,0)}(G)$-function
$g(W_{0},W_{1},W_{2})$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime},W_{2}^{\prime})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v\\}$ and $W_{2}^{\prime}=W_{2}\times\\{v\\}$,
is a $(1,0,0)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}\setminus W_{0}\times V(H)$, we can see that
$g^{\prime}_{(x,v)\rightarrow(x,y)}$ is a $(1,0,0)$-dominating function on
$G\circ H$. Furthermore, for every $x\in W_{0}$ there exists $x^{\prime}\in
W_{2}\cap N(x)$ or two vertices $x^{\prime},x^{\prime\prime}\in W_{1}\cap
N(x)$, and so $g^{\prime}_{(x^{\prime},v)\rightarrow(x,y)}$ is a
$(1,0,0)$-dominating function on $G\circ H$ for every $(x,y)\in W_{0}\times
V(H)$. Therefore, $g^{\prime}$ is a secure $(1,0,0)$-dominating function on
$G\circ H$, and as a consequence, $\gamma_{(1,0,0)}^{s}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,1,0)}(G)$. ∎

We next present some particular cases of Theorem 3.11.

######  Corollary 3.12.

Let $n$ and $r$ be two integers such that $n\geq r\geq 2$. If $H$ is a
noncomplete graph with $\gamma(H)=1$, then the following statements hold.

  * •

$\gamma_{(1,0,0)}^{s}(K_{n}\circ H)=2$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{1,n-1}\circ H)=2$.

  * •

$\gamma_{(1,0,0)}^{s}(K_{n,r}\circ H)=\left\\{\begin{array}[]{ll}3&\text{if
}r=2,\\\\[5.0pt] 4&\mbox{otherwise.}\end{array}\right.$

###### Proof.

Theorem 3.11 leads to the result, as
$\gamma_{(2,1,0)}(K_{n})=\gamma_{(2,1,0)}(K_{1,n-1})=2$,
$\gamma_{(2,1,0)}(K_{n,2})=3$ and $\gamma_{(2,1,0)}(K_{n,r})=4$ whenever
$r\geq 3$. ∎

To conclude this section, we would present the following result which shows
that the study of the cases $G\cong P_{n}$ and $G\cong C_{n}$ is complete.

######  Theorem 3.13.

[18] For any integer $n\geq 3$ and any noncomplete graph with $\gamma(H)=1$,

$\gamma_{(1,0,0)}^{s}(P_{n}\circ
H)=2\left\lceil\frac{n}{3}\right\rceil\quad\text{and}\quad\gamma_{(1,0,0)}^{s}(C_{n}\circ
H)=\left\lceil\frac{2n}{3}\right\rceil.$

###  3.3 Secure total domination and total weak Roman domination

It was shown in [9] that for any graph $G$ with no isolated vertex and any
nontrivial graph $H$ the secure total domination number of $G\circ H$ equals
the total weak Roman domination number.

######  Theorem 3.14.

[9] For any graph $G$ with no isolated vertex and any nontrivial graph $H$,

$\gamma_{(1,1,1)}^{s}(G\circ H)=\gamma_{(1,1)}^{s}(G\circ H).$

According to Theorem 3.14, we can restrict ourselves to study the secure total
domination number of $G\circ H$. To this end, we shall need the following
lemma.

######  Lemma 3.15.

[9] For any graph $G$ with no isolated vertex and any nontrivial graph $H$,
there exists a $\gamma_{(1,1)}^{s}(G\circ H)$-function $f$ satisfying that
$f(V(H_{u}))\leq 2$ for every $u\in V(G)$.

The following result shows that the secure total domination number of $G\circ
H$ equals $\gamma_{w}^{s}(G)$ or $\gamma_{w}(G)$ for certain vectors $w$ of
three components. As we can expect, the decision on whether the components of
$w$ take specific values will depend on the value of $\gamma_{(1,1)}^{s}(H)$
and/or $\gamma(H)$.

######  Theorem 3.16.

For a graph $G$ with no isolated vertex and a nontrivial graph $H$, the
following statements hold.

  1. (i)

If $\gamma_{(1,1)}^{s}(H)=2$, then $\gamma_{(1,1)}^{s}(G\circ
H)=\gamma_{(1,1,0)}^{s}(G)$.

  2. (ii)

If $\gamma(H)=1$ and $\gamma_{(1,1)}^{s}(H)\geq 3$, then
$\gamma_{(1,1)}^{s}(G\circ H)=\gamma_{(1,1,1)}^{s}(G)$.

  3. (iii)

If $\gamma(H)=2<\gamma_{(1,1)}^{s}(H)$, then $\gamma_{(1,1)}^{s}(G\circ
H)=\gamma_{(2,2,1)}(G)$.

  4. (iv)

If $\gamma(H)\geq 3$, then $\gamma_{(1,1)}^{s}(G\circ H)=\gamma_{(2,2,2)}(G)$.

###### Proof.

Let $f(V_{0},V_{1})$ be a $\gamma_{(1,1)}^{s}(G\circ H)$-function which
satisfies Lemma 3.15. Let $f^{\prime}(X_{0},X_{1},X_{2})$ be the function
defined on $G$ by $X_{1}=\\{x\in V(G):f(V(H_{x}))=1\\}$ and $X_{2}=\\{x\in
V(G):f(V(H_{x}))=2\\}$. With this notation in mind, we differentiate the
following four cases.

Case $1.$ $\gamma_{(1,1)}^{s}(H)=2$. If $x\in X_{0}\cup X_{1}$, then
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$, which implies that
$f^{\prime}$ is a $(1,1,0)$-dominating function on $G$. Now, for any $(x,y)\in
X_{0}\times V(H)$, there exists $(x^{\prime},y^{\prime})\in M_{f}(x,y)$ with
$x^{\prime}\in N(x)\cap(X_{1}\cup X_{2})$. Hence, for any $u\in X_{0}\cup
X_{1}\cup\\{x^{\prime}\\}$ we have that $f^{\prime}_{x^{\prime}\rightarrow
x}(N(u))=f_{(x^{\prime},y^{\prime})\rightarrow(x,y)}(N(V(H_{u}))\setminus
V(H_{u}))\geq 1$, which implies that $f^{\prime}$ is a secure
$(1,1,0)$-dominating function on $G$. Therefore, $\gamma_{(1,1)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(1,1,0)}^{s}(G)$.

On the other side, for any $\gamma_{(1,1,0)}^{s}(G)$-function
$g(W_{0},W_{1},W_{2})$ and any two universal vertices $v_{1},v_{2}$ of $H$,
the function $g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=(W_{1}\times\\{v_{1}\\})\cup(W_{2}\times\\{v_{1},v_{2}\\})$,
is a $(1,1)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}\setminus W_{0}\times V(H)$, we can see that
$g^{\prime}_{(x,v_{1})\rightarrow(x,y)}$ is a $(1,1)$-dominating function on
$G\circ H$. Furthermore, for every $x\in W_{0}$ there exists $x^{\prime}\in
M_{g}(x)$, and so $g^{\prime}_{(x^{\prime},v_{1})\rightarrow(x,y)}$ is a
$(1,1)$-dominating function on $G\circ H$ for every $(x,y)\in W_{0}\times
V(H)$. Therefore, $\gamma_{(1,1)}^{s}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(1,1,0)}^{s}(G)$.

Case $2.$ $\gamma(H)=1$ and $\gamma_{(1,1)}^{s}(H)\geq 3$. Since
$f(V(H_{x})\leq 2$ and $\gamma_{(1,1)}^{s}(H_{x})\geq 3$ for any $x\in V(G)$,
we have that $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 1$. Thus,
$f^{\prime}$ is a $(1,1,1)$-dominating function on $G$. Now, for any $(x,y)\in
X_{0}\times V(H)$, there exists $(x^{\prime},y^{\prime})\in M_{f}(x,y)$ with
$x^{\prime}\in N(x)\cap(X_{1}\cup X_{2})$. Hence, for any $u\in V(G)$ we have
that $f^{\prime}_{x^{\prime}\rightarrow
x}(N(u))=f_{(x^{\prime},y^{\prime})\rightarrow(x,y)}(N(V(H_{u}))\setminus
V(H_{u}))\geq 1$, which implies that $f^{\prime}$ is a secure
$(1,1,1)$-dominating function on $G$. Therefore, $\gamma_{(1,1)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(1,1,1)}^{s}(G)$.

Now we show that $\gamma_{(1,1)}^{s}(G\circ H)\leq\gamma_{(1,1,1)}^{s}(G)$.
Let $v$ be the universal vertex of $H$, which is unique, as
$\gamma_{(1,1)}^{s}(H)\geq 3$. For any $\gamma_{(1,1,1)}^{s}(G)$-function
$g(W_{0},W_{1},W_{2})$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime},W^{\prime}_{2})$, defined by
$W_{1}^{\prime}=W_{1}\times\\{v\\}$ and $W_{2}^{\prime}=W_{2}\times\\{v\\}$,
is a $(1,1,1)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}\setminus W_{0}\times V(H)$, we can see that
$g^{\prime}_{(x,v)\rightarrow(x,y)}$ is a $(1,1,1)$-dominating function on
$G\circ H$. Furthermore, for every $x\in W_{0}$ there exists $x^{\prime}\in
M_{g}(x)$, and so $g^{\prime}_{(x^{\prime},v)\rightarrow(x,y)}$ is a
$(1,1,1)$-dominating function on $G\circ H$ for every $(x,y)\in W_{0}\times
V(H)$. Therefore, $g^{\prime}$ is a secure $(1,1,1)$-dominating function on
$G\circ H$, and by Theorem 3.14 we deduce that $\gamma_{(1,1)}^{s}(G\circ
H)=\gamma_{(1,1,1)}^{s}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(1,1,1)}^{s}(G)$.

Case $3.$ $\gamma(H)=2<\gamma_{(1,1)}^{s}(H)$. Assume first that $x\in
X_{0}\cup X_{1}$. Since $\gamma_{(1,1)}^{s}(H_{x})\geq 3$, we have that
$f^{\prime}(N(x))=f(N(V(H_{x}))\setminus V(H_{x}))\geq 2$. Analogously, for
any $x\in X_{2}$ we deduce that $f^{\prime}(N(x))=f(N(V(H_{x}))\setminus
V(H_{x}))\geq 1$. Therefore, $f^{\prime}$ is a $(2,2,1)$-dominating function
on $G$ and, as a consequence, $\gamma_{(1,1)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,2,1)}(G)$.

On the other side, for any $\gamma_{(2,2,1)}(G)$-function
$g(W_{0},W_{1},W_{2})$ and any $\gamma(H)$-set $\\{v_{1},v_{2}\\}$, the
function $g^{\prime}(W_{0}^{\prime},W_{1}^{\prime})$, defined by
$W_{1}^{\prime}=(W_{1}\times\\{v_{1}\\})\cup(W_{2}\times\\{v_{1},v_{2}\\})$,
is a $(1,1)$-dominating function on $G\circ H$. Now, for any $(x,y)\in
W_{0}^{\prime}\setminus W_{2}\times V(H)$ and $x^{\prime}\in
N(x)\cap(W_{1}\cup W_{2})$ we have that
$g^{\prime}_{(x^{\prime},v_{1})\rightarrow(x,y)}(N(u,v))\geq 1$ for every
$(u,v)\in V(G\circ H)$. Moreover, for any $(x,y)\in
W_{0}^{\prime}\cap(W_{2}\times V(H))$ we have that
$g^{\prime}_{(x,v_{1})\rightarrow(x,y)}(N(u,v))\geq 1$ or
$g^{\prime}_{(x,v_{2})\rightarrow(x,y)}(N(u,v))\geq 1$ for every $(u,v)\in
V(G\circ H)$. Therefore, $g^{\prime}$ is a secure $(1,1)$-dominating function
on $G\circ H$, which implies that $\gamma_{(1,1)}^{s}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,2,1)}(G)$.

Case $4.$ $\gamma(H)\geq 3$. In this case, for every $x\in V(G)$, there exists
$y\in V(H)$ such that $f(N[(x,y)]\cap V(H_{x}))=0$. Hence,
$f^{\prime}(N(x))=f(N(x,y)\setminus V(H_{x}))\geq 2$ for every $x\in V(G)$.
Therefore, $f^{\prime}$ is a $(2,2,2)$-dominating function on $G$, and so
$\gamma_{(1,1)}^{s}(G\circ
H)=\omega(f)=\omega(f^{\prime})\geq\gamma_{(2,2,2)}(G)$.

Now, for any $\gamma_{(2,2,2)}(G)$-function $g(W_{0},W_{1},W_{2})$ and any
$v\in V(H)$, the function
$g^{\prime}(W_{0}^{\prime},W_{1}^{\prime},W_{2}^{\prime})$, defined by
$W_{2}^{\prime}=W_{2}\times\\{v\\}$ and $W_{1}^{\prime}=W_{1}\times\\{v\\}$,
is a secure $(2,2,2)$-dominating function on $G\circ H$. Therefore,
$\gamma_{(2,2,2)}(G\circ H)\leq\omega(g^{\prime})$, and by Theorems 2.2 and
3.14 we deduce that $\gamma_{(1,1)}^{s}(G\circ H)=\gamma_{(1,1,1)}^{s}(G\circ
H)\leq\gamma_{(2,2,2)}(G\circ
H)\leq\omega(g^{\prime})=\omega(g)=\gamma_{(2,2,2)}(G)$. According to the
cases above, the result follows. ∎

The following result is a particular case of Theorem 3.16.

######  Corollary 3.17.

The following statements hold for a nontrivial graph $H$ and any integers
$n,r$ such that $n\geq r\geq 2$.

  * •

$\gamma_{(1,1)}^{s}(K_{n}\circ H)=\left\\{\begin{array}[]{ll}2&\text{if
}\,\gamma(H)=1,\\\\[5.0pt] 3&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,1)}^{s}(K_{1,n-1}\circ H)=\left\\{\begin{array}[]{ll}2&\text{if
}\,\gamma_{(1,1)}^{s}(H)=2,\\\\[5.0pt] 4&\text{if }\,\gamma(H)\geq
3,\\\\[5.0pt] 3&\mbox{otherwise.}\end{array}\right.$

  * •

$\gamma_{(1,1)}^{s}(K_{n,r}\circ H)=\left\\{\begin{array}[]{ll}3&\text{if
}\,\gamma(H)=1\text{ and }r=2,\\\\[5.0pt]
4&\mbox{otherwise.}\end{array}\right.$

###### Proof.

The result follows from Theorem 3.16 by considering the following facts.

  * •

$\gamma_{(1,1,0)}^{s}(K_{n})=\gamma_{(1,1,1)}^{s}(K_{n})=2$ and
$\gamma_{(2,2,1)}(K_{n})=\gamma_{(2,2,2)}(K_{n})=3$.

  * •

$\gamma_{(1,1,0)}^{s}(K_{1,n-1})=2$,
$\gamma_{(1,1,1)}^{s}(K_{1,n-1})=\gamma_{(2,2,1)}(K_{1,n-1})=3$ and
$\gamma_{(2,2,2)}(K_{1,n-1})=4$.

  * •

If $r=2$, then
$\gamma_{(1,1,0)}^{s}(K_{n,r})=\gamma_{(1,1,1)}^{s}(K_{n,r})=3$, while if
$r\geq 3$, then
$\gamma_{(1,1,0)}^{s}(K_{n,r})=\gamma_{(1,1,1)}^{s}(K_{n,r})=\gamma_{(2,2,1)}(K_{n,r})=\gamma_{(2,2,2)}(K_{n,r})=4$.

∎

By combining Theorem 3.16 and some known results we derive the following
results.

######  Theorem 3.18.

The following statements hold for any integer $n\geq 4$ and any nontrivial
graph $H$.

  1. (i)

If $\gamma_{(1,1)}^{s}(H)=2$, then $\gamma_{(1,1)}^{s}(P_{n}\circ
H)=2\left\lceil\frac{n}{3}\right\rceil$.

  2. (ii)

If $\gamma(H)=1$ and $\gamma_{(1,1)}^{s}(H)\geq 3$, then
$\gamma_{(1,1)}^{s}(P_{n}\circ
H)=\gamma_{(1,1,1)}^{s}(P_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{TWRDF2018}{}{}]}}}}{{=}}\left\lceil\frac{5(n-2)}{7}\right\rceil+2.$

  3. (iii)

If $\gamma(H)=2<\gamma_{(1,1)}^{s}(H)$, then $\gamma_{(1,1)}^{s}(P_{n}\circ
H)=\gamma_{(2,2,1)}(P_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{w-domination}{}{}]}}}}{{=}}\left\\{\begin{array}[]{ll}n-\lfloor\frac{n}{7}\rfloor+1&\text{if
}\,n\equiv 1,2\pmod{7},\\\\[5.0pt]
n-\lfloor\frac{n}{7}\rfloor&\mbox{otherwise.}\end{array}\right.$

  4. (iv)

If $\gamma(H)\geq 3$, then $\gamma_{(1,1)}^{s}(P_{n}\circ
H)=\gamma_{(2,2,2)}(P_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{w-domination}{}{}]}}}}{{=}}\left\\{\begin{array}[]{ll}n&if\,n\equiv
0\pmod{4},\\\\[5.0pt] n+1&if\,n\equiv 1,3\pmod{4},\\\\[5.0pt] n+2&if\,n\equiv
2\pmod{4}.\end{array}\right.$

###### Proof.

As indicated in the statements, we only need to prove (i). In this case, by
Theorem 3.16 we know that $\gamma_{(1,1)}^{s}(P_{n}\circ
H)=\gamma_{(1,1,0)}^{s}(P_{n})$. Now, by Theorem 2.1 (ii) and Corollary 2.3 we
deduce that
$\gamma_{(2,1,0)}(P_{n})\leq\gamma_{(1,1,0)}^{s}(P_{n})\leq\gamma_{(2,2,0)}(P_{n})$.
Moreover, as shown in [4],
$\gamma_{(2,1,0)}(P_{n})=\gamma_{(2,2,0)}(P_{n})=2\left\lceil\frac{n}{3}\right\rceil$,
which completes the proof. ∎

By the result above and Theorem 3.16 we deduce the following result.

######  Proposition 3.19.

For any integer $n\geq 4$,

$\gamma_{(1,1,0)}^{s}(P_{n})=2\left\lceil\frac{n}{3}\right\rceil.$

The following result concerns the case when $G$ is a cycle.

######  Theorem 3.20.

The following statements hold for any integer $n\geq 4$ and any nontrivial
graph $H$.

  1. (i)

If $\gamma_{(1,1)}^{s}(H)=2$, then $\gamma_{(1,1)}^{s}(C_{n}\circ
H)=\gamma_{(1,1,0)}^{s}(C_{n})=\left\\{\begin{array}[]{ll}\left\lceil\frac{2n}{3}\right\rceil&if\,n=4,7,\\\\[5.0pt]
2\left\lceil\frac{n}{3}\right\rceil&\text{otherwise}.\end{array}\right.$

  2. (ii)

If $\gamma(H)=1$ and $\gamma_{(1,1)}^{s}(H)\geq 3$, then
$\gamma_{(1,1)}^{s}(C_{n}\circ
H)=\gamma_{(1,1,1)}^{s}(C_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{TWRDF2018}{}{}]}}}}{{=}}\left\lceil\frac{5n}{7}\right\rceil.$

  3. (iii)

If $\gamma(H)=2<\gamma_{(1,1)}^{s}(H)$, then

$\gamma_{(1,1)}^{s}(C_{n}\circ
H)=\gamma_{(2,2,1)}(C_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{w-domination}{}{}]}}}}{{=}}\left\\{\begin{array}[]{ll}n-\lfloor\frac{n}{7}\rfloor+1&\text{if
}\,n\equiv 1,2\pmod{7},\\\\[5.0pt]
n-\lfloor\frac{n}{7}\rfloor&\mbox{otherwise.}\end{array}\right.$

  4. (iv)

If $\gamma(H)\geq 3$, then $\gamma_{(1,1)}^{s}(C_{n}\circ
H)=\gamma_{(2,2,2)}(C_{n})\stackrel{{\scriptstyle\mbox{\rm\cite[cite]{[\@@bibref{}{w-domination}{}{}]}}}}{{=}}n.$

###### Proof.

As indicated in the statements, we only need to prove (i). In this case, by
Theorem 3.16 we know that $\gamma_{(1,1)}^{s}(C_{n}\circ
H)=\gamma_{(1,1,0)}^{s}(C_{n})$. As shown in [4],
$\gamma_{(2,1,0)}(C_{n})=\left\lceil\frac{2n}{3}\right\rceil$ and
$\gamma_{(2,2,0)}(C_{n})=2\left\lceil\frac{n}{3}\right\rceil$. Moreover, by
Theorem 2.1 (ii) and Corollary 2.3 we deduce that
$\gamma_{(2,1,0)}(C_{n})\leq\gamma_{(1,1,0)}^{s}(C_{n})\leq\gamma_{(2,2,0)}(C_{n})$.
Therefore, $\gamma_{(1,1,0)}^{s}(C_{n})\leq
2\left\lceil\frac{n}{3}\right\rceil$ and, if $n\equiv 0,2\pmod{3}$, then
$\gamma_{(1,1,0)}^{s}(C_{n})=2\left\lceil\frac{n}{3}\right\rceil$.

From now on, we consider that $n\equiv 1\pmod{3}$ with $n\geq 10$, as the
cases $n=4$ and $n=7$ are very easy to check. Let $f(V_{0},V_{1},V_{2})$ be a
$\gamma_{(1,1,0)}^{s}(C_{n})$-function and let
$V(C_{n})=\\{x_{0},\ldots,x_{n-1}\\}$, where consecutive vertices are adjacent
and the addition of subscripts is taken modulo $n$. If there exists $x_{i}\in
V(C_{n})$ such that $f(x_{i})=f(x_{i+1})=0$, then
$\gamma_{(1,1,0)}^{s}(C_{n})=\gamma_{(1,1,0)}^{s}(P_{n})$ and we derived the
result by Proposition 3.19. Hence, we assume that for any $x_{i}\in V_{0}$ we
have that $f(x_{i-1})>0$ and $f(x_{i+1})>0$. From this fact, and considering
that $f$ is a secure $(1,1,0)$-dominating function, we deduce that
$f(\\{x_{j},x_{j+1},x_{j+2}\\})\geq 2$ for any $x_{j}\in V(C_{n})$. Now, we
consider the following two cases.

Case $1$. $V_{2}\neq\emptyset$. Without loss of generality, we suppose that
$x_{0}\in V_{2}$ and let $S_{i}=\\{x_{3i+1},x_{3i+2},x_{3i+3}\\}$ for
$i\in\\{0,\ldots,\frac{n-4}{3}\\}$. Notice that $f(S_{i})\geq 2$ for every
$i\in\\{0,\ldots,\frac{n-4}{3}\\}$, which implies that

$\gamma_{(1,1,0)}^{s}(C_{n})=\omega(f)\geq\sum_{i=0}^{\frac{n-4}{3}}f(S_{i})+f(x_{0})\geq\frac{2(n-1)}{3}+2=2\left\lceil\frac{n}{3}\right\rceil.$

This implies that
$\gamma_{(1,1,0)}^{s}(C_{n})=2\left\lceil\frac{n}{3}\right\rceil.$

Case $2$. $V_{2}=\emptyset$. In this case, if there exist four consecutive
vertices with weight one, namely without loss of generality
$x_{1},x_{2},x_{3},x_{4}$, then we deduce that

$\gamma_{(1,1,0)}^{s}(C_{n})=\omega(f)\geq\sum_{i=5}^{n}f(x_{i})+f(\\{x_{1},x_{2},x_{3},x_{4}\\})\geq\frac{2(n-4)}{3}+4=2\left\lceil\frac{n}{3}\right\rceil.$

From now on we assume that for any group of four consecutive vertices, at
least one vertex has weight equal to zero. Moreover, notice that $|V_{0}|\geq
2$ as $n\geq 10$ and $\gamma_{(1,1,0)}^{s}(C_{n})\leq
2\left\lceil\frac{n}{3}\right\rceil.$ Without loss of generality, let
$x_{0}\in V_{0}$. and suppose that $x_{1}\in M_{f}(x_{0})$. This implies that
$f(x_{1})=f(x_{2})=f(x_{3})=1$ and $f(x_{4})=0$, and so $f(x_{5})=f(x_{6})=1$.

If $f(x_{7})=1$, then we have that

$\gamma_{(1,1,0)}^{s}(C_{n})=\omega(f)\geq\sum_{i=8}^{n}f(x_{i})+f(\\{x_{1},x_{2},\ldots,x_{7}\\})\geq\frac{2(n-7)}{3}+6=2\left\lceil\frac{n}{3}\right\rceil,$

Now, assume that $f(x_{7})=0$. Thus, $n\geq 13$ and $x_{8}\in M_{f}(x_{7})$,
which implies that $f(x_{8})=f(x_{9})=f(x_{10})=1$. Hence,

$\gamma_{(1,1,0)}^{s}(C_{n})=\omega(f)\geq\sum_{i=11}^{n}f(x_{i})+f(\\{x_{1},x_{2},\ldots,x_{10}\\})\geq\frac{2(n-10)}{3}+8=2\left\lceil\frac{n}{3}\right\rceil,$

which completes the proof. ∎

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