# Operations on Fuzzy Incidence Graphs and Strong Incidence Domination

Kavya. R. Nair 111Corresponding author. Kavya. R. Nair, Department of
Mathematics, National Institute of Technology, Calicut, Kerala, India. Email:
<EMAIL_ADDRESS>and M. S. Sunitha<EMAIL_ADDRESS>
Department of Mathematics, National Institute of Technology, Calicut, Kerala,
India-673601

###### Abstract

Fuzzy incidence graphs (FIG) model real world problems efficiently when there
is an extra attribute of vertex- edge relationship. The article discusses the
operations on Fuzzy incidence graphs. The join, Cartesian product, tensor
product, and composition of FIGs are explored. The study is concentrated
mainly on strong fuzzy incidence graphs (SFIG). The idea of strong incidence
domination (SID) is used, and strong incidence domination number (SIDN) in
operations is examined. Basic properties of FIGs obtained from the operations
are studied. Bounds for the domination number of product of two SFIGs are
determined for the Cartesian and tensor products. Study is conducted on FIGs
with strong join and composition. Complete fuzzy incidence graphs (CFIGs) and
FIGs with effective pairs are also considered in the study.

Keywords: Fuzzy incidence graphs, Weak fuzzy incidence cycle, Cartesian
product, join, tensor product, composition, strong fuzzy incidence graphs.

##  1 Introduction

With the introduction of fuzzy set theory by Zadeh [1] in 1965, the theory has
evolved in various ways across numerous fields. The theory has wide range of
applications in operation research, pattern recognition, decision theory,
artificial intelligence etc. Graph-theoretical principles are frequently
employed in the research and modeling of diverse applications in various
fields. However, in many circumstances, graph-theoretical notions are
ambiguous and imprecise. In such cases it is suitable to use fuzzy set
approaches to cope with ambiguity and uncertainty. This led to the
introduction of fuzzy graph (FG) theory by Rosenfeld [2] in 1975. The
introduction of fuzzy incidence graphs (FIG) was motivated by a FG model with
the extra attribute of vertices having some effect on the edges. Dinesh [3]
developed the term FIG in 2016 and studied some of its properties. The idea of
connectivity and fuzzy end nodes is developed by Mordeson et al. [4, 5]. It is
often customary to study and perform operations on graph structures to obtain
new structures from the existing ones. Since the 1950s, a number of graph
products have been investigated. The operations on fuzzy graphs are discussed
by Mordeson and Peng [6]. Parvathi et al. [7] examined several intuitionistic
fuzzy graph (IFG) operations such as Cartesian product and composition. Sahoo
and Pal [8] explored direct product, strong product and semistrong product in
IFGs. Nazeer et al. studied the intuitionistic fuzzy incidence graphs (IFIG)
products in [9]. The research on domination theory in graphs and fuzzy graphs
expanded over the years due to its applications in numerous fields. The
concept of domination emerged in the 1850s with the chessboard domination
problem. Ore [10] and Berge [11] pioneered the study of domination in graphs
in 1962. Significant works on fuzzy graph domination and domination in fuzzy
graph products have been done in [14, 13, 16, 17, 12, 15]. Nazeer et al. [18,
19] proposed FIG domination based on effective pairs and defined strong
domination in FIG and their join. Afsharmanesh et al. [20] recently proposed
domination using valid edges in FIGs. Kavya and Sunitha [21] defined strong
incidence domination using weight of strong pairs.
The study is motivated by the fact that the pairs in every FIG may be
characterised as strong or non-strong. Hence the idea of SID developed can
always be applied in any FIG. Furthermore, unlike other domination parameters
that use the weight of vertices, SID uses the weight of strong pairs. As a
result, the SID yields the lowest value. Also, graph operations are always
beneficial for generating new structures from the already known structures.
Hence the study on operations on FIG is conducted, and the idea of SID is
applied to the obtained graphs.
The article studies some of the operations on FIGs. The study mainly deals
with the join, Cartesian product, tensor product, and composition of SFIGs.
Section 1 sums up the preliminaries. Section 2 discusses the join of FIGs and
SID in the join. A characterisation for a FIG to be a SFIG is proved. It is
illustrated with an example that the join of two SFIG need not be strong in
general. A sufficient condition for the join to be strong is considered.
Section 3 and 4 deal with the Cartesian and tensor products of SFIGs and FIGs
with effective pairs respectively. Section 4 studies the composition of SFIGs.
A sufficient condition for the composition to be strong is proved. Bound for
the SIDN is obtained in each of the sections.

##  2 Preliminaries

The following definitions are taken from [3, 4, 5, 21, 18, 19, 9]. Throughout
the article minimum and maximum operators are represented by $\wedge$ and
$\vee$ respectively. A triple $X=(V,E,I)$ such that $V$ is non-empty,
$E\subseteq V\times V\text{ and }I\subseteq V\times E$ is an incidence graph
(IG).
Elements in I are called incidence pairs or pairs and are of the form $(x,yz)$
where $x\in V$ and $e^{\prime}=yz\in E$.
If $(w,xw),(x,xw),(y,yz)\text{ and }(z,yz)$ are in $I$, then the edges $wx$
and $yz$ are considered adjacent.
An incidence subgraph, $Y$ of $X$ is an IG having all its vertices, edges and
pairs in $X$.
A sequence of vertices, edges and pairs starting at $x^{\prime}$ and ending at
$y^{\prime}$, where $x^{\prime},y^{\prime}\in V\cup E$ is called an incidence
walk from $x^{\prime}$ to $y^{\prime}$. An incidence trail is an incidence
walk with distinct pairs. If the vertices in an incidence walk are distinct ,
then it is called an incidence path. If each vertex in an IG is joined to
every other vertex by a path, then the IG is connected. A component of an IG
is a maximally connected incidence subgraph of the IG.
Let $X=(V,E)$ be a graph. Let $\varepsilon$ and $\rho$ be fuzzy subsets of $V$
and $E\subseteq V\times V$ respectively. Then
$\mathscr{X}=(V,\varepsilon,\rho)$ is fuzzy graph(FG) of $X$ if
$\rho(xy)\leq\varepsilon(x)\wedge\varepsilon(y)$ for all $x,y\in V$. Also, if
$\eta(v^{\prime},e^{\prime})\leq\varepsilon(v^{\prime})\wedge\rho(e^{\prime})$,
for all $v^{\prime}\in V\text{ and }e^{\prime}\in E$, then $\eta$ is the fuzzy
incidence of $X$. And, $\mathscr{\tilde{X}}=(\varepsilon,\rho,\eta)$ is called
fuzzy incidence graph (FIG) of $X$.
Here, $\varepsilon^{*},\rho^{*}\text{ and }\eta^{*}$ are defined as
$\varepsilon^{*}=\\{x\in V:\varepsilon(x)>0\\}$, $\rho^{*}=\\{e^{\prime}\in
E:\rho(e^{\prime})>0\\}$, and $\eta^{*}=\\{(x,xy)\in I:\eta(x,xy)>0\\}$. If
$|\varepsilon^{*}|=1$, then the FIG is called trivial.
Also, $\mathscr{X}=(V,\varepsilon,\rho)$ and
$\mathscr{X}^{*}=(V,\varepsilon^{*},\rho^{*})$ are the underlying FG of
$\mathscr{\tilde{X}}$ and underlying graph of $\mathscr{X}$ respectively. Let
$xy\in\rho^{*}$, if $(x,xy),(y,xy)\in\eta^{*}$, then $xy$ is an edge in
$\tilde{\mathscr{X}}$. Pairs in FIG, $\mathscr{\tilde{X}}$ are elements of the
form $(x,xy)$. If vertices $x$ and $y$ are joined by a path, then $x$ and $y$
are connected in $\tilde{\mathscr{X}}$. If each vertex is joined to every
other vertex by a path, then $\tilde{\mathscr{X}}$ is connected. A fuzzy
incidence subgraph $\tilde{\mathscr{Y}}=(\varphi,\xi,\zeta)$ of
$\tilde{\mathscr{X}}$ is such that
$\varphi\subseteq\varepsilon,\xi\subseteq\rho,\text{ and }\zeta\subseteq\eta$
and $\tilde{\mathscr{Y}}$ is a fuzzy incidence spanning subgraph of
$\mathscr{\tilde{X}}$ if $\varphi=\varepsilon$. If
$\varphi=\varepsilon,\xi=\rho,\text{ and }\zeta=\eta$ for elements in
$\varphi^{*},\xi^{*},\zeta^{*}$ respectively, then $\tilde{\mathscr{Y}}$ is a
subgraph of $\mathscr{\tilde{X}}$.
A FIG, $\tilde{\mathscr{X}}$ is complete fuzzy incidence graph (CFIG) if
$\eta(x,xy)=\,\wedge\\{\varepsilon(x),\rho(xy)\\}$ for all $(x,xy)\in V\times
E$ and $\rho(xy)=\varepsilon(x)\wedge\varepsilon(y)$ for all $(x,y)\in V\times
V$. A pair $(x,xy)$ is an effective pair if
$\eta(x,xy)=\,\wedge\\{\varepsilon(x),\rho(xy)\\}$.
In a FIG $\tilde{\mathscr{X}}$, a path from $s^{\prime}$ to $t^{\prime}$ where
$s^{\prime},t^{\prime}\in\varepsilon^{*}\cup\rho^{*}$, is called an incidence
path. The minimum of the $\eta$ values of pairs in an incidence path is the
incidence strength of that path. Here, $\eta^{\infty}(x,yz)\text{ or
}ICONN_{\tilde{\mathscr{X}}}(x,yz)$ is denoted as the incidence strength of
path from $x$ to $yz$ of greatest incidence strength.
If $\tilde{\mathscr{X^{*}}}=(\varepsilon^{*},\rho^{*},\eta^{*})$ is a cycle,
then $\tilde{\mathscr{X}}$ is a cycle. In addition, $\tilde{\mathscr{X}}$ is a
fuzzy cycle (FC), if there exists no unique edge $xy\in\rho^{*}$ with the
least weight. A FIG, $\tilde{\mathscr{X}}=(\varepsilon,\rho,\eta)$ is a fuzzy
incidence cycle (FIC) if $\tilde{\mathscr{X}}$ is a FC and there is no unique
$(x,xy)\in\eta^{*}$ with the least weight.
Let $\tilde{\mathscr{X}}=(\varepsilon,\rho,\eta)$ be a FIG,
$\eta^{\prime\infty}(x,yz)$ is the greatest incidence strength among the
incidence strength of all paths from $x$ to $yz$ in
$\tilde{\mathscr{G}}\setminus(x,yz)$. If
$\eta(x,xy)>\eta^{\prime\infty}(x,xy)$, then the pair $(x,xy)$ is $\alpha-$
strong. Pair $(x,xy)$ is $\beta-$ strong if
$\eta(x,xy)=\eta^{\prime\infty}(x,xy)$, and is a $\delta-$ pair if
$\eta(x,xy)<\eta^{\prime\infty}(x,xy)$. A strong pair is $\alpha-$ strong or
$\beta-$ strong pair. Vertex $x$ and edge $xy$ are strong fuzzy incidence
neighbors if $(x,xy)$ is strong. A path with strong pairs is called strong
incidence path (SIP). A FIG with only strong pairs is called a strong fuzzy
incidence graph (SFIG).
If both $(x,xy)$ and $(y,xy)$ are strong, then $x$ is called strong incidence
neighbor (SIN) of $y$. Also, $N_{IS}(x)$ is the strong incidence neighborhood
of $x$, which is the set of all SINs of $x$. If $x=y$ or $x$ is a SIN of $y$,
then $x$ dominates $y$. Isolated vertex $x$ is such that $N_{IS}(x)=\phi$. A
set $\tilde{\mathscr{D}}\subseteq V$ in $\tilde{\mathscr{X}}$ is a strong
incidence dominating set (SIDS), if for any $x\in V-\tilde{\mathscr{D}}$,
$\exists$ some $y\in\tilde{\mathscr{D}}$ such that, $x$ is a SIN of $y$.
Here, $W(\tilde{\mathscr{D}})$ is the weight of SIDS, $\tilde{\mathscr{D}}$,
defined as

$W(\tilde{\mathscr{D}})=\sum_{x\in\tilde{\mathscr{D}}}\eta(x,xy)$

where $\eta(x,xy)$ is minimum weight of strong pairs at $x$ and $y\in
N_{IS}(x)$. The strong incidence domination number (SIDN), denoted as
$\gamma_{IS}(\tilde{\mathscr{X}})$ or $\gamma_{IS}$ is the minimum weight of
the SIDSs in the FIG, $\tilde{\mathscr{X}}$. A minimum SIDS is a SIDS with
minimum weight.

##  3 Strong incidence domination in Join of Fuzzy Incidence Graph

This section studies the SID in join of FIGs. The section begins with the
definition of a WFIC. Theorem 3.4 is a characterisation for a FIG to be
strong. It is established that the join of two SFIG need not be strong.
Proposition 3.8 proves a necessary condition for the join of two FIGs to be
strong. A sufficient condition for the join of two FIGs to be strong is also
proved in Theorem 3.10. Theorem 3.13 and its corollaries discuss the SIDN and
SIDS in the join of FIGs.

######  Remark 3.1.

The definition of weak fuzzy incidence cycle (WFIC) is given in Definition
3.2. A WFIC is different from a FIC, since it is not necessary for the
underlying FG of a WFIC to be a FC. An example of a WFIC is illustrated in
Example 3.3.

######  Definition 3.2.

Let $\tilde{\mathscr{X}}=(\varepsilon,\rho,\eta)$ be a FIG such that
$(\varepsilon^{*},\rho^{*},\eta^{*})$ is a cycle and there is no unique
$(a,ab)\in\eta^{*}$ such that
$\eta(a,ab)=\wedge\\{\eta(c,cd)|(c,cd)\in\eta^{*}\\}$. Then
$\tilde{\mathscr{X}}$ is called a weak fuzzy incidence cycle (WFIC).

$y(1)$$v(1)$$w(1)$$x(1)$$u(1)$0.30.20.30.50.10.30.30.20.20.30.30.50.50.10.1

Fig. 1. Weak fuzzy incidence cycle $\tilde{\mathscr{X}}$

######  Example 3.3.

The FIG, $\tilde{\mathscr{X}}$ in Fig. 1. is an example of WFIC, but
$\tilde{\mathscr{X}}$ is not a FIC, since the underlying FG is not a FC.

######  Theorem 3.4.

A FIG, $\tilde{\mathscr{X}}$ is a SFIG iff every cycle in
$\tilde{\mathscr{X}}$ is a WFIC.

###### Proof.

Let $\tilde{\mathscr{X}}$ be a FIG such that every cycle in
$\tilde{\mathscr{X}}$ is a WFIC. Let $(a,ab)$ be a pair in
$\tilde{\mathscr{X}}$. Then there are 3 cases:
Case 1: $(a,ab)$ does not belongs to any cycle of $\tilde{\mathscr{X}}$.
Then $(a,ab)$ is a strong pair.
Case 2: In every cycle that contains $(a,ab)$, $(a,ab)$ is a pair with least
weight.
Since every cycle in $\tilde{\mathscr{X}}$ is a WFIC this implies that
$\eta(a,ab)=ICONN_{\tilde{\mathscr{X}}\setminus(a,ab)}(a,ab)$, i.e, $(a,ab)$
is a strong pair.
Case 3: There exists at least one cycle $\tilde{C}$ containing $(a,ab)$, in
which $(a,ab)$ is not a pair with least weight.
Then $\eta(a,ab)$ is greater than the incidence strength of the path
$\tilde{C}\setminus(a,ab)$, which implies that $\eta(a,ab)\geq
ICONN_{\tilde{\mathscr{X}}\setminus(a,ab)}(a,ab)$ i.e, $(a,ab)$ is a strong
pair.
Since $(a,ab)$ is arbitrary, it implies that $\tilde{\mathscr{X}}$ is a SFIG.
Conversely, suppose that $\tilde{\mathscr{X}}$ is a SFIG. Suppose that
$\tilde{C}$ is a cycle that is not WFIC. Then there exists a pair $(a,ab)$ in
$\tilde{C}$ such that $(a,ab)$ is the unique weakest pair in $\tilde{C}$. The
path $\tilde{C}\setminus(a,ab)$ has incidence strength greater than
$\eta(a,ab)$. Hence
$\eta(a,ab)<ICONN_{\tilde{\mathscr{X}}\setminus(a,ab)}(a,ab)$, i.e, $(a,ab)$
is a $\delta-$ pair, which contradicts the assumption. Hence the result. ∎

The definition of join of FIGs is taken from [19].

######  Definition 3.5.

[19] Let $\tilde{\mathscr{X}_{1}}=(\varepsilon_{1},\rho_{1},\eta_{1})$ and
$\tilde{\mathscr{X}_{2}}=(\varepsilon_{2},\rho_{2},\eta_{2})$ be two FIGs.
Then the join of $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$
denoted as $\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is the FIG,
$\tilde{\mathscr{X}}=(\varepsilon,\rho,\eta)$ such that:

$\varepsilon(a)=\begin{cases}\varepsilon_{1}(a)&if\quad
a\in\tilde{\mathscr{X}_{1}}\\\ \varepsilon_{2}(a)&if\quad
a\in\tilde{\mathscr{X}_{2}}\end{cases}$
$\rho(ab)=\begin{cases}\rho_{1}(ab)&if\quad a,b\in\tilde{\mathscr{X}_{1}}\\\
\rho_{2}(ab)&if\quad a,b\in\tilde{\mathscr{X}_{2}}\\\
\varepsilon_{1}(a)\wedge\varepsilon_{2}(b)&if\quad
a\in\tilde{\mathscr{X}_{1}}\text{ and }b\in\tilde{\mathscr{X}_{2}}\end{cases}$

$\eta(a,ab)=\begin{cases}\eta_{1}(a,ab)&if\quad(a,ab)\in\tilde{\mathscr{X}_{1}}\\\
\eta_{2}(a,ab)&if\quad(a,ab)\in\tilde{\mathscr{X}_{2}}\\\
\varepsilon_{1}(a)\wedge\varepsilon_{2}(b)\wedge\eta_{1}(a,av_{i})&if\quad
a\in\tilde{\mathscr{X}_{1}}\text{ and }b\in\tilde{\mathscr{X}_{2}}\\\
&\qquad\text{ where }v_{i}\in\tilde{\mathscr{X}_{1}}\\\
\varepsilon_{1}(b)\wedge\varepsilon_{2}(a)\wedge\eta_{2}(a,av_{i})&if\quad
b\in\tilde{\mathscr{X}_{1}}\text{ and }a\in\tilde{\mathscr{X}_{2}}\\\
&\qquad\text{ where }v_{i}\in\tilde{\mathscr{X}_{2}}\end{cases}$

######  Remark 3.6.

In general, the join of two SFIGs need not be strong. Example 3.7 illustrates
that join of two SFIGs need not be SFIG.

######  Example 3.7.

For the FIGs in Fig. 2. $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ are SFIGs. But the join
$\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is not a SFIG. The pair
$(y,xy)$ is $\delta-$ pair since
$ICONN_{\tilde{\mathscr{X}}-(y,xy)}=0.1>\eta(y,xy)=0.05$ Also, by Theorem 3.4,
there exists a cycle $xyzux$ with unique weakest pair. Hence
$\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is not strong.

Next, Proposition 3.8 gives a necessary condition for the join of two FIGs to
be strong.

######  Proposition 3.8.

If $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are two FIGs such
that the join
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is
strong, then $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are
SFIGs.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs such
that the join
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is
strong. Suppose that $\tilde{\mathscr{X}_{1}}$ is not strong. Then there
exists a pair $(a,ab)$ such that
$\eta(a,ab)<ICONN_{\tilde{\mathscr{X}_{1}}\setminus(a,ab)}(a,ab)$, i.e,
$\eta(a,ab)$ is less than the incidence strength of a path say, $P$ in
$\tilde{\mathscr{X}_{1}}\setminus(a,ab)$. Then the path $P$ exists in the join
$\tilde{\mathscr{X}}$ also. Since the incidence strength between $a$ and $ab$
is the maximum of incidence strength of all paths between $a$ and $ab$,
$\eta(a,ab)<ICONN_{\tilde{\mathscr{X}}\setminus(a,ab)}(a,ab)$. Hence $(a,ab)$
is not strong in join which is a contradiction. Therefore,
$\tilde{\mathscr{X}_{1}}$ is SFIG. By the same argument it can be proved that
$\tilde{\mathscr{X}_{2}}$ is a SFIG. ∎

######  Proposition 3.9.

Let $\tilde{\mathscr{X}}$ be a FIG such that for each vertex $x$ in
$\tilde{\mathscr{X}}$, every pair incident at $x$ has equal weight, then
$\tilde{\mathscr{X}}$ is a SFIG.

###### Proof.

Suppose $\tilde{\mathscr{X}}$ is a FIG such that for each vertex $x$ in
$\tilde{\mathscr{X}}$, every pair incident at $x$ has equal weight. Let
$(a,ab)$ be an arbitrary pair in $\tilde{\mathscr{X}}$. Now, every FIP in
$\tilde{\mathscr{X}}\setminus(a,ab)$ from $a$ to $ab$ begins with a pair
incident at $a$. Since every pair incident at $a$ has equal weight, it implies
that $\eta(a,ab)=\eta(a,ac)\geq
ICONN_{\tilde{\mathscr{X}}\setminus(a,ab)}(a,ab)$. Hence $(a,ab)$ is a strong
pair. Since $(a,ab)$ is arbitrary, $\tilde{\mathscr{X}}$ is a SFIG. ∎

Theorem 3.10 gives a sufficient condition for the join of two FIGs,
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ to be strong.

######  Theorem 3.10.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be FIGs such that
for each vertex $u$ in $\tilde{\mathscr{X}_{i}}$, $i=1,2$, every pair incident
at $u$ has the same weight. Then the join of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ is a SFIG.

###### Proof.

Let
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$,
and $a$ be a vertex in $\tilde{\mathscr{X}_{1}}$. Then, every pair in
$\tilde{\mathscr{X}_{1}}$ incident at $a$ has equal weight say w. Now, for any
vertex $b\in\tilde{\mathscr{X}_{2}}$ the pair $(a,ab)$ has weight

$\begin{split}\eta(a,ab)&=\varepsilon_{1}(a)\wedge\varepsilon_{2}(b)\wedge\eta(a,at_{i}),t_{i}\in\tilde{\mathscr{X}_{1}}\\\
&=\varepsilon_{2}(b)\wedge\text{w}\end{split}$

Therefore, every pair incident at $a$ in $\tilde{\mathscr{X}}$ will have
weight less than or equal to w.
Hence for any pair $(a,ac)$, $a\in$ $\tilde{\mathscr{X}_{1}}$ there are two
cases;
Case 1: $\eta(a,ac)=$ w.
Any incidence path from $a$ to $ac$ in $\tilde{\mathscr{X}}\setminus(a,ac)$
starts with pair incident at $a$. Since every pair at $a$ has weight either w
or less than w, it follows that

$ICONN_{\tilde{\mathscr{X}}\setminus(a,ac)}(a,ac)\leq\text{w}=\eta(a,ac)$

$\implies$ $(a,ac)$ is strong pair.
Case 2: $\eta(a,ac)=\varepsilon_{2}(c)$
By the definition of join of FIGs, $\eta(c,ac)\leq\varepsilon_{2}(c)$.
Therefore as in Case 1, any incidence path from $a$ to $ac$ in
$\tilde{\mathscr{X}}\setminus(a,ac)$ ends with pair $(c,ac)$. Hence, it
follows that,

$ICONN_{\tilde{\mathscr{X}}\setminus(a,ac)}(a,ac)\leq\eta(c,ac)\leq\varepsilon_{2}(c)=\eta(a,ac)$

$\implies$ $(a,ac)$ is strong pair.
Similarly it can be proved for the pairs incident at vertices in
$\tilde{\mathscr{X}_{2}}$. Hence it implies that $\tilde{\mathscr{X}}$ is
SFIG. ∎

######  Remark 3.11.

If two SFIGs have property as in Theorem 3.10, then their join is also a SFIG.
The converse of Theorem 3.10 need not be true as illustrated in Example 3.12.

######  Example 3.12.

The FIGs, $\tilde{\mathscr{X}_{1}}$, $\tilde{\mathscr{X}_{2}}$ and the join
$\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ in Fig. 3. are strong,
but the pairs incident at $v$ in $\tilde{\mathscr{X}_{1}}$ have weights 0.1
and 0.2 and hence the converse of Theorem 3.10 need not be true.

######  Theorem 3.13.

If $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are FIGs such that
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is
strong, then

$\gamma_{IS}=\wedge\\{W(D_{1}),W(D_{2}),W(\\{a,b\\})\text{ where
}a\in\tilde{\mathscr{X}_{1}}\text{ and }b\in\tilde{\mathscr{X}_{2}}\\}$

where, $D_{1}$ and $D_{2}$ are minimum SIDS of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively.

###### Proof.

Since $\tilde{\mathscr{X}}$ is strong, every pair in $\tilde{\mathscr{X}}$ is
strong pair. Hence each vertex in $\tilde{\mathscr{X}_{1}}$ dominates every
vertex in $\tilde{\mathscr{X}_{2}}$ and vice versa. Therefore $\\{a,b\\}$
where $a\in\tilde{\mathscr{X}_{1}}$ and $b\in\tilde{\mathscr{X}_{2}}$
dominates every vertex in $\tilde{\mathscr{X}}$, and hence is a SIDS. In a
similar way if $D_{1}$ is a minimum SIDS of $\tilde{\mathscr{X}_{1}}$, it
dominates every vertex of $\tilde{\mathscr{X}_{1}}$ and any vertex in $D_{1}$
dominates every vertex in $\tilde{\mathscr{X}_{2}}$ in the join. Hence $D_{1}$
is a SIDS of $\tilde{\mathscr{X}}$. By the same argument, $D_{2}$ is also a
SIDS. Hence the result. ∎

######  Corollary 3.14.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are FIGs such that
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}}$ is
strong. If $D$ is the minimum SIDS of $\tilde{\mathscr{X}}$ then
$|D|=min\\{|D_{1}|,|D_{2}|,2\\}$ where, $D_{1}$ and $D_{2}$ are minimum SIDS
of $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively.

######  Corollary 3.15.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be FIGs such that
for each vertex $u$ in $\tilde{\mathscr{X}_{i}}$, $i=1,2$, every pair incident
at $u$ has the same weight. Then,
$\gamma_{IS}(\tilde{\mathscr{X}_{1}}\oplus\tilde{\mathscr{X}_{2}})=min\\{W(D_{1}),W(D_{2}),W(\\{a,b\\})\text{
where }a\in\tilde{\mathscr{X}_{1}}\text{ and }b\in\tilde{\mathscr{X}_{2}}\\}$
where, $D_{1}$ and $D_{2}$ are minimum SIDS of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively.

##  4 Strong incidence domination in Cartesian Product of Fuzzy Incidence
Graph

The section discusses the Cartesian product of FIGs, SFIGs and FIGs with
effective pairs. Theorem 4.4 proves that the Cartesian product of two SFIGs is
strong. Proposition 4.6 discusses the Cartesian product in FIGs with effective
pairs. Proposition 4.12 establishes the Cartesian product and the SID in
CFIGs. The SID in the Cartesian product of SFIGs is also discussed in Theorem
4.13.

######  Remark 4.1.

The operations on IFIGs is defined and studied in [9]. In this article
definition 4.2, 5.1 and 6.1 are modified from the definitions in [9].

######  Definition 4.2.

Let
$\tilde{\mathscr{X}_{1}}=(V_{1},E_{1},I_{1},\varepsilon_{1},\rho_{1},\eta_{1})$
and
$\tilde{\mathscr{X}_{2}}=(V_{2},E_{2},I_{2},\varepsilon_{2},\rho_{2},\eta_{2})$
be two FIGs. Then the Cartesian product of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ denoted as
$\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$ is the FIG,
$\tilde{\mathscr{X}}=(V,E,I,\varepsilon,\rho,\eta)$ such that: $V=V_{1}\times
V_{2}$, $E=\\{(a_{1},b_{1})(a_{2},b_{2})|a_{1}=a_{2},\,b_{1}b_{2}\in
E_{2}\text{ \, or \, }a_{1}a_{2}\in E_{1},\,b_{1}=b_{2}\\}$ and
$I=\\{((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))|a_{1}=a_{2},(b_{1},b_{1}b_{2})\in
I_{2},(b_{2},b_{1}b_{2})\in I_{2}\text{ \,or\,
}b_{1}=b_{2},(a_{1},a_{1}a_{2})\in I_{1},(a_{2},a_{1}a_{2})\in I_{1}\\}$ and

$\varepsilon(a_{1},a_{2})=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(a_{2})\quad\forall(a_{1},a_{2})\in
V_{1}\times V_{2}$
$\rho((a_{1},b_{1})(a_{2},b_{2}))=\begin{cases}\varepsilon_{1}(a_{1})\wedge\rho_{2}(b_{1}b_{2})&if\quad
a_{1}=a_{2},b_{1}b_{2}\in E_{2}\\\
\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\quad
b_{1}=b_{2},a_{1}a_{2}\in E_{1}\end{cases}$
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\begin{cases}\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})&if\quad
a_{1}=a_{2},\\\ &\qquad(b_{1},b_{1}b_{2})\in I_{2}\\\
\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\quad b_{1}=b_{2},\\\
&\qquad(a_{1},a_{1}a_{2})\in I_{1}\end{cases}$

######  Remark 4.3.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two CFIGs with
$m$ and $n$ vertices respectively. Let $a_{1},a_{2},...,a_{m}$ be the vertices
of $\tilde{\mathscr{X}_{1}}$ and $b_{1},b_{2},...,b_{n}$ be the vertices of
$\tilde{\mathscr{X}_{2}}$. Consider the vertices of
$\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$ in the form of rows and
columns as follows,
$(a_{1},b_{1}),(a_{1},b_{2}),(a_{1},b_{3}),...,(a_{1},b_{n})$
$(a_{2},b_{1}),(a_{2},b_{2}),(a_{2},b_{3}),...,(a_{2},b_{n})$
$\vdots$
$(a_{m},b_{1}),(a_{m},b_{2}),(a_{m},b_{3}),...,(a_{m},b_{n})$
Each vertex is adjacent to all vertices in the same row and column in which it
belongs. Since $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are
CFIGs, they are SFIGs. Hence, each vertex dominates all vertices in the same
row and column in which it belongs.

######  Theorem 4.4.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs, then
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$ is
a SFIG.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs and
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$ be
the Cartesian product of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$. Consider a pair say,
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ in $\tilde{\mathscr{X}}$. Then,
either $a_{1}=a_{2},(b_{1},b_{1}b_{2}),(b_{2},b_{1}b_{2})\in I_{2}$ or
$(a_{1},a_{1}a_{2}),(a_{2},a_{1}a_{2})\in I_{1},b_{1}=b_{2}$.
Consider the case when $a_{1}=a_{2},(b_{1},b_{1}b_{2}),(b_{2},b_{1}b_{2})\in
I_{2}$. Then,
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})$.
Case 1:
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\varepsilon_{1}(a_{1})$
Then, the pair weight

$\displaystyle\eta((a_{1},b_{2}),(a_{1},b_{2})(a_{1},b_{1}))$
$\displaystyle=\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{2},b_{1}b_{2})$
$\displaystyle\leq\varepsilon_{1}(a_{1})$

Any path from $(a_{1},b_{1})$ to $(a_{1},b_{1})(a_{1},b_{2})$ contains the
pair $((a_{1},b_{2}),(a_{1},b_{2})(a_{1},b_{1}))$. Since the weight of
$((a_{1},b_{2}),(a_{1},b_{2})(a_{1},b_{1}))$ is less than or equal to
$\varepsilon_{1}(a_{1})$, $((a_{1},b_{1}),(a_{1},b_{1})(a_{1},b_{2}))$ is a
strong pair.
Case 2:
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\eta_{2}(b_{1},b_{1}b_{2})$.
Now, consider paths from $(a_{1},b_{1})$ to $(a_{1},b_{2})$.
Sub case 1: Consider a path from $(a_{1},b_{1})$ to $(a_{1},b_{2})$ such that
the vertices in the path are of the form $(a_{1},v)$, $v\in V_{2}$.
Corresponding to each such path, say,
$P:(a_{1},b_{1}),(a_{1},v_{1}),(a_{1},v_{2}),...,(a_{1},v_{n}),(a_{1},b_{2})$,
$v_{1},v_{2},...,v_{n}\in\tilde{\mathscr{X}_{2}}$ and
$b_{1}v_{1},v_{1}v_{2},...,v_{n}b_{2}\in E_{2}$, there exists a path
$P^{\prime}:b_{1},v_{1},v_{2},...,v_{n},b_{2}$ in $\tilde{\mathscr{X}_{2}}$.
The path $P^{\prime}$ together with
$\\{(b_{2},b_{1}b_{2}),b_{1}b_{2},(b_{1},b_{1}b_{2})\\}$ forms a cycle, say
$C$. Since $\tilde{\mathscr{X}_{2}}$ is a SFIG every cycle contains more than
one weakest pair. Therefore, in $C$, there exists a pair with weight
$\leq\eta_{2}(b_{1},b_{1}b_{2})$. Since
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})=\eta(b_{1},b_{1}b_{2})$,
and the weight of each pair in path $P$ is defined as
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}v_{1})$,
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{1},b_{1}v_{1})$,
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{1},v_{1}v_{2})$,
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{2},v_{1}v_{2})$ ,…,
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{n},v_{n}b_{2})$,
$\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{2},v_{n}b_{2})$, it implies that the
path
$P\cup\\{((a_{1},b_{2}),(a_{1},b_{1})(a_{1},b_{2})),(a_{1},b_{1})(a_{1},b_{2})\\}$
has strength $\leq\eta_{2}(b_{1},b_{1}b_{2})$.
Sub case 2: Consider a path from $(a_{1},b_{1})$ to $(a_{1},b_{2})$ that
contains vertices of the form $(a_{n},b_{1})$ and $(a_{n},b_{2})$, $a_{n}\neq
a_{1}$.
In each such path there will be an edge of the form
$(a_{i},b_{1})(a_{i},b_{2})$ with pair weights
$\eta((a_{i},b_{1}),(a_{i},b_{1})(a_{i},b_{2}))=\varepsilon_{1}(a_{i})\wedge\eta_{2}(b_{1},b_{1}b_{2})$
and
$\eta((a_{i},b_{2}),(a_{i},b_{1})(a_{i},b_{2}))=\varepsilon_{1}(a_{i})\wedge\eta_{2}(b_{2},b_{1}b_{2})$.
Hence, the path contains pair with weight $\leq\eta_{2}(b_{1},b_{1}b_{2}).$
Therefore, the strength of the path is $\leq\eta(b_{1},b_{1}b_{2}).$
Sub case 3: Consider a path from $(a_{1},b_{1})$ to $(a_{1},b_{2})$ that
contains vertices of the form $(a_{n},b_{m})$, $b_{m}\neq b_{1},b_{2}$. Then
corresponding to each such path, say, $P$ in $\tilde{\mathscr{X}}$, from
$(a_{1},b_{1})$ to $(a_{1},b_{2})$, there exists a path in
$\tilde{\mathscr{X}_{2}}$ from $b_{1}$ to $b_{2}$ other than the edge
$b_{1}b_{2}$. This path in $\tilde{\mathscr{X}_{2}}$ together with
$\\{(b_{2},b_{1}b_{2}),b_{1}b_{2},(b_{1},b_{1}b_{1})\\}$ forms a cycle, say,
$C$ in $\tilde{\mathscr{X}_{2}}$. Since $\tilde{\mathscr{X}_{2}}$ is a SFIG,
$C$ has more than one weakest pair. And hence
$P\cup\\{(a_{1},b_{1})(a_{1},b_{2}),((a_{1},b_{2}),(a_{1},b_{1})(a_{1},b_{2}))\\}$
in $\tilde{\mathscr{X}}$ has a pair of weight
$\leq\eta_{2}(b_{1},b_{1}b_{2})$. Hence the pair
$((a_{1},b_{1}),(a_{1},b_{1})(a_{1},b_{2}))$ is strong. The case when
$b_{1}=b_{2},(a_{1},a_{1}a_{2}),(a_{2},a_{1}a_{2})\in I_{1}$ can be proved
similarly by taking $\tilde{\mathscr{X}_{1}}$ as SFIG. Therefore, if
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are strong, then
$\tilde{\mathscr{X}}$ is a SFIG. ∎

######  Remark 4.5.

By the Definition 4.2, of Cartesian product of FIGs, $\tilde{\mathscr{X}_{1}}$
and $\tilde{\mathscr{X}_{2}}$, a pair of non- adjacent vertices always exists
in $\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$.
Therefore, the Cartesian product of two FIGs can never be CFIG.

######  Proposition 4.6.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs with
effective pairs, then
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$ is
a FIG with effective pairs.

###### Proof.

Let $\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$
be the Cartesian product of two FIGs $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ with effective pairs. Consider a pair
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ in $\tilde{\mathscr{X}}$, and the
pair is effective if
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\varepsilon(a_{1},b_{1})\wedge\rho((a_{1},b_{1})(a_{2},b_{2}))$.
Since $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are FIGs with
effective pairs and by Definition 4.2, the weight of
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is

$\displaystyle\eta((a_{1},b_{1}),($ $\displaystyle a_{1},b_{1})(a_{2},b_{2}))$
$\displaystyle=\begin{cases}\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})&if\quad
a_{1}=a_{2},\quad(b_{1},b_{1}b_{2})\in I_{2}\\\
\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\quad
b_{1}=b_{2},\quad(a_{1},a_{1}a_{2})\in I_{1}\end{cases}$
$\displaystyle=\begin{cases}\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\rho_{2}(b_{1}b_{2})&if\quad
a_{1}=a_{2},\\\ &\qquad(b_{1},b_{1}b_{2})\in I_{2}\\\
\varepsilon_{1}(a_{1})\wedge\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\quad
b_{1}=b_{2},\\\ &\qquad(a_{1},a_{1}a_{2})\in I_{1}\end{cases}$
$\displaystyle=\begin{cases}\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\varepsilon_{1}(a_{1})\wedge\rho_{2}(b_{1}b_{2})&if\quad
a_{1}=a_{2},\\\ &\qquad(b_{1},b_{1}b_{2})\in I_{2}\\\
\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\quad
b_{1}=b_{2},\\\ &\qquad(a_{1},a_{1}a_{2})\in I_{1}\end{cases}$
$\displaystyle=\varepsilon(a_{1},b_{1})\wedge\rho((a_{1},b_{1})(a_{2},b_{2}))$

which implies that the pair $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is
effective. Hence, $\tilde{\mathscr{X}}$ is a FIG with effective pairs. ∎

######  Theorem 4.7.

[4] Let $\tilde{\mathscr{X}}=(\varepsilon,\rho,\eta)$ be a FIG. If
$\eta(x,xy)=\wedge\\{\varepsilon(x),\rho(xy)\\}$, then pair $(x,xy)$ is
strong.

######  Proposition 4.8.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs such
that there exits an edge $a_{1}a_{2}\in E_{1}$ with pair weights
$\eta(a_{1},a_{1}a_{2}),\eta(a_{2},a_{1}a_{2})\geq\varepsilon_{2}(v)$ for some
$v\in V_{2}$, then the pairs $((a_{1},v),(a_{1},v)(a_{2},v))$ and
$((a_{2},v),(a_{1},v)(a_{2},v))$ are effective pairs.

###### Proof.

Suppose $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are two FIGs
such that there exits an edge $a_{1}a_{2}\in E_{1}$ with pair weights
$\eta(a_{1},a_{1}a_{2}),\eta(a_{2},a_{1}a_{2})\geq\varepsilon_{2}(v)$ for some
$v\in V_{2}$. Since $\rho_{1}(a_{1}a_{2})\geq\eta(a_{1},a_{1}a_{2})$ and
$\rho_{1}(a_{1}a_{2})\geq\eta(a_{2},a_{1}a_{2})$, the weight of edge
$(a_{1},v)(a_{2},v)$ in $E$ is
$\rho((a_{1},v)(a_{2},v))=\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(v)=\varepsilon_{2}(v)$.
Therefore,
$\eta((a_{1},v),(a_{1},v)(a_{2},v))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(v)=\varepsilon_{2}(v)=\rho((a_{1},v)(a_{2},v))\wedge\varepsilon_{2}(v)$.
Similarly,
$\eta((a_{2},v),(a_{1},v)(a_{2},v))=\eta_{1}(a_{2},a_{1}a_{2})\wedge\varepsilon_{2}(v)=\varepsilon_{2}(v)=\rho((a_{1},v)(a_{2},v))\wedge\varepsilon_{2}(v)$.
Hence, the pairs
$((a_{1},v),(a_{1},v)(a_{2},v))$ and $((a_{2},v),(a_{1},v)(a_{2},v))$ are
effective pairs. ∎

Corollary 4.9 follows from Theorem 4.7 and Proposition 4.8.

######  Corollary 4.9.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs such
that there exits an edge $a_{1}a_{2}\in E_{1}$ with pair weights
$\eta(a_{1},a_{1}a_{2}),\eta(a_{2},a_{1}a_{2})\geq\varepsilon_{2}(v)$ for some
$v\in V_{2}$, then the pairs $((a_{1},v),(a_{1},v)(a_{2},v))$ and
$((a_{2},v),(a_{1},v)(a_{2},v))$ are strong pairs.

######  Proposition 4.10.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs. If
the pairs $(a_{1},a_{1}a_{2})$ and $(a_{2},a_{1}a_{2})$ are effective pairs in
$\tilde{\mathscr{X}_{1}}$, then for any $v\in V_{2}$, the pairs
$((a_{1},v),(a_{1},v)(a_{2},v))$ and $((a_{2},v),(a_{1},v)(a_{2},v))$ are
effective in $\tilde{\mathscr{X}}$.

###### Proof.

Consider two FIGs, $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$.
Since $(a_{1},a_{1}a_{2})$ and $(a_{2},a_{1}a_{2})$ are effective pairs in
$\tilde{\mathscr{X}_{1}}$,
$\eta_{1}(a_{1},a_{1}a_{2})=\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{1}(a_{1})=\rho_{1}(a_{1}a_{2})$
and
$\eta_{1}(a_{2},a_{1}a_{2})=\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{1}(a_{2})=\rho_{1}(a_{1}a_{2})$.
Therefore,
$\eta((a_{1},v),(a_{1},v)(a_{2},v))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(v)=\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(v)=\rho((a_{1},v)(a_{2},v))\wedge\varepsilon_{2}(v)$
and
$\eta((a_{2},v),(a_{1},v)(a_{2},v))=\eta_{1}(a_{2},a_{1}a_{2})\wedge\varepsilon_{2}(v)=\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(v)=\rho((a_{1},v)(a_{2},v))\wedge\varepsilon_{2}(v)$.
Hence the pairs $((a_{1},v),(a_{1},v)(a_{2},v))$ and
$((a_{2},v),(a_{1},v)(a_{2},v))$ are effective in $\tilde{\mathscr{X}}$. ∎

######  Corollary 4.11.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs. If
the pairs $(a_{1},a_{1}a_{2})$ and $(a_{2},a_{1}a_{2})$ are effective pairs in
$\tilde{\mathscr{X}_{1}}$, then for any $v\in V_{2}$, the pairs
$((a_{1},v),(a_{1},v)(a_{2},v))$ and $((a_{2},v),(a_{1},v)(a_{2},v))$ are
strong in $\tilde{\mathscr{X}}$.

######  Proposition 4.12.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two CFIGs with
$m$ and $n$ vertices respectively. Let $D$ be a minimum dominating set of
$\tilde{\mathscr{X}_{1}}\times\tilde{\mathscr{X}_{2}}$, then
$|D|=\wedge\\{m,n\\}$ and $\gamma_{IS}=\wedge\\{m,n\\}\times$w where w is the
weight of vertex having the least weight in $V_{1}\cup V_{2}$.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two CFIGs with
$m$ and $n$ vertices respectively. Let $a_{1},a_{2},...,a_{m}$ be the vertices
of $\tilde{\mathscr{X}_{1}}$ and $b_{1},b_{2},...,b_{n}$ be the vertices of
$\tilde{\mathscr{X}_{2}}$. Suppose that $m<n$. Then,
$D_{1}=\\{(a_{1},b_{1}),(a_{2},b_{2}),....,(a_{m},b_{m})\\}$ will dominate all
vertices of $\tilde{\mathscr{X}}$. This is because, $(a_{1},b_{1})$ dominates
the vertices in the set $\\{(a_{1},b_{2}),(a_{1},b_{3}),...,(a_{1},b_{n})\\}$,
$(a_{2},b_{2})$ dominates the vertices in
$\\{(a_{2},b_{1}),(a_{2},b_{3}),...,(a_{2},b_{n})\\}$ and so on. Finally,
$(a_{m},b_{m})$ dominates vertices in
$\\{(a_{m},b_{1}),(a_{m},b_{2}),...,(a_{m},b_{n})\\}$. Therefore all vertices
in $\tilde{\mathscr{X}}$ are dominated by $m$ vertices in $D_{1}$.
Case 1: Vertex with least weight belongs to $\tilde{\mathscr{X}_{1}}.$
Without loss of generality assume that $a_{1}$ is a vertex in
$\tilde{\mathscr{X}_{1}}$ with least weight. Then the pairs incident at
$a_{1}$ has weight $\varepsilon_{1}(a_{1})$. Now, consider the dominating set
$D_{1}$. The first vertex $(a_{1},b_{1})$ is adjacent to $(a_{1},b_{2})$,
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{1},b_{2}))=\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\rho_{2}(b_{1}b_{2})=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\varepsilon_{2}(b_{2})$=$\varepsilon_{1}(a_{1})$
i.e, the minimum of the weight of the pairs at $(a_{1},b_{1})$ is
$\varepsilon_{1}(a_{1})$. Similarly, $(a_{2},b_{2})$ is adjacent to
$(a_{1},b_{2})$ and the weight of the pair
$((a_{2},b_{2}),(a_{2},b_{2})(a_{1},b_{2}))$ is $\varepsilon_{1}(a_{1})$ and
so on. In the same way $(a_{m},b_{m})$ is adjacent to $(a_{1},b_{m})$ and the
minimum of the weight of the pairs at $(a_{m},b_{m})$ is
$\varepsilon_{1}(a_{1})$. Therefore, the weight of $D_{1}$ is
$m\varepsilon_{1}(a_{1})$.
Case 2: Vertex with least weight belongs to $\tilde{\mathscr{X}_{2}}.$
Without loss of generality assume that $b_{1}$ is a vertex in
$\tilde{\mathscr{X}_{2}}$ with least weight. Then the pairs incident at
$b_{1}$ has weight $\varepsilon_{2}(b_{1})$. Now consider the dominating set
$D_{1}$. The first vertex $(a_{1},b_{1})$ is adjacent to $(a_{2},b_{1})$, and
the weight of the pair $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{1}))$ is
$\varepsilon_{2}(b_{1})$, i.e, the minimum of the weight of the pairs at
$(a_{1},b_{1})$ is $\varepsilon_{2}(b_{1})$. Similarly, $(a_{2},b_{2})$ is
adjacent to $(a_{2},b_{1})$ and the weight of the pair
$((a_{2},b_{2}),(a_{2},b_{2})(a_{2},b_{1}))$ is $\varepsilon_{2}(b_{1})$ and
so on. In the same way $(a_{m},b_{m})$ is adjacent to $(a_{m},b_{1})$ and the
minimum of the weight of the pairs at $(a_{m},b_{m})$ is
$\varepsilon_{2}(b_{1})$. Therefore, the weight of $D_{1}$ is
$m\varepsilon_{2}(b_{1})$.
Among the dominating sets with $m$ vertices, the least weight is obtained for
$D_{1}$ which is $m$w, where w is the weight of vertex in
$\tilde{\mathscr{X}_{1}}$ or $\tilde{\mathscr{X}_{2}}$ having the least
weight. Now, assume $D_{2}=\\{a_{1},a_{2},...,a_{m-1}\\}$ is a dominating set
of $\tilde{\mathscr{X}_{1}}$ with $m-1$ vertices. Since there are $m$ rows of
vertices in $\tilde{\mathscr{X}}$, there exists a row or column of vertices
which is not dominated by any of the $m-1$ vertices.
Hence, the minimum dominating set contains $m$ vertices and $\gamma_{IS}=m$w
where w is the vertex having the least weight in $V_{1}\cup V_{2}$.
The case of $n<m$ and $n=m$ can be proved in a similar way and in that case
the minimum dominating set contains $n$ vertices and $\gamma_{IS}=n$w where w
is the weight of vertex having the least weight.
Hence the result. ∎

######  Theorem 4.13.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs, and
$D_{1}$ and $D_{2}$ are SIDS of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively. Then $D_{1}\times V2$ and $V_{1}\times
D_{2}$ are SIDS of $\tilde{\mathscr{X}}$ and

$\gamma_{IS}\leq\wedge\\{W(D_{1}\times V_{2}),W(V_{1}\times D_{2})\\}.$

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs, and
$D_{1}$ and $D_{2}$ are SIDS of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively. By Theorem 4.4, $\tilde{\mathscr{X}}$
is a SFIG.
Let $(u,v)$ be any vertex in $\tilde{\mathscr{X}}$. There are two cases:
Case 1: At least one of $u$ or $v$ or both are isolated.
Sub case 1: Vertex $u$ is isolated in $\tilde{\mathscr{X}_{1}}$.
Since $D_{1}$ is a SIDS, $u\in D_{1}$. If $v\in D_{2}$, then $(u,v)\in
D_{1}\times V_{2}$ and $(u,v)\in V_{1}\times D_{2}$. And if $v\notin D_{2}$,
then there exists a vertex $v^{\prime}\in D_{2}$ such that $v^{\prime}$
dominates $v$ in $\tilde{\mathscr{X}_{2}}$. Hence, $(u,v^{\prime})\in
V_{1}\times D_{2}$ dominates $(u,v)$ in $\tilde{\mathscr{X}}$. Also, $(u,v)\in
D_{1}\times V_{2}$.
Sub case 2: Vertex $v$ is isolated in $\tilde{\mathscr{X}_{2}}$.
Since $D_{2}$ is a SIDS, $v\in D_{2}$. If $u\in D_{1}$, then $(u,v)\in
D_{1}\times V_{2}$ and $(u,v)\in V_{1}\times D_{2}$. And if $u\notin D_{1}$,
then there exists a vertex $u^{\prime}\in D_{1}$ such that $u^{\prime}$
dominates $u$ in $\tilde{\mathscr{X}_{1}}$. Hence, $(u^{\prime},v)\in
D_{1}\times V_{2}$ dominates $(u,v)$ in $\tilde{\mathscr{X}}$. Also, $(u,v)\in
V_{1}\times D_{2}$.
Sub case 3: Both vertices $u$ and $v$ are isolated in
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively.
Then $(u,v)\in D_{1}\times V_{2}$ and $(u,v)\in V_{1}\times D_{2}$.
Hence if at least one of $u$ and $v$ is an isolated vertex in
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively, then
$(u,v)$ either belongs to $D_{1}\times V_{2}$ and $V_{1}\times D_{2}$ or is
dominated by vertex in $D_{1}\times V_{2}$ and $V_{1}\times D_{2}$.
Case 2: Both $u$ and $v$ are non isolated.
Consider $D_{1}\times V_{2}$, and if $(u,v)\notin D_{1}\times V_{2}$, then
$u\notin D_{1}$ and since $D_{1}$ is a SIDS of $\tilde{\mathscr{X}_{1}}$,
there exists $u^{\prime}\in D_{1}$ such that $(u,uu^{\prime})$ and
$(u^{\prime},uu^{\prime})$ are strong pairs in $\tilde{\mathscr{X}_{1}}$,
i.e., $u^{\prime}$ dominates $u$ in $\tilde{\mathscr{X}_{1}}$. Then, by the
definition of Cartesian product of FIGs $(u,v)$ and $(u^{\prime},v)$ are
adjacent. Therefore, $(u,v)(u^{\prime},v)\in E$ and
$((u,v),(u,v)(u^{\prime},v))$ and $((u^{\prime},v),(u,v)(u^{\prime},v))$ are
strong pairs in $\tilde{\mathscr{X}}$, i.e., $(u^{\prime},v)$ dominates
$(u,v)$. Hence $D_{1}\times V_{2}$ is a SIDS in $\tilde{\mathscr{X}}$.
Similarly, $V_{1}\times D_{2}$ is also a SIDS in $\tilde{\mathscr{X}}$.
Therefore, clearly $\gamma_{IS}\leq\wedge\\{W(D_{1}\times V_{2}),W(V_{1}\times
D_{2})\\}$. ∎

##  5 Strong Incidence Domination in Tensor Product of Fuzzy Incidence Graphs

The section discusses the tensor product of FIGs, SFIGs and FIGs with
effective pairs. Theorem 5.2 proves that the tensor product of FIGs with
effective pairs is a FIG with effective pairs. Theorem 5.5 establishes that
the tensor product of two SFIGs is a SFIG. Theorem 5.6 discusses the SID in
the tensor product of two SFIG.

######  Definition 5.1.

Let
$\tilde{\mathscr{X}_{1}}=(V_{1},E_{1},I_{1},\varepsilon_{1},\rho_{1},\eta_{1})$
and
$\tilde{\mathscr{X}_{2}}=(V_{2},E_{2},I_{2},\varepsilon_{2},\rho_{2},\eta_{2})$
be two FIGs. Then the tensor product of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ denoted as
$\tilde{\mathscr{X}_{1}}\diamond\tilde{\mathscr{X}_{2}}$ is the FIG,
$\tilde{\mathscr{X}}=(V,E,I,\varepsilon,\rho,\eta)$ such that: $V=V_{1}\times
V_{2}$, $E=\\{(a_{1},b_{1})(a_{2},b_{2})|\quad a_{1}a_{2}\in E_{1}\text{ and
}b_{1}b_{2}\in E_{2}\\}$ and
$I=\\{((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))|\quad(a_{1},a_{1}a_{2}),(a_{2},a_{1}a_{2})\in
I_{1}\text{ and }(b_{1},b_{1}b_{2}),(b_{2},b_{1}b_{2})\in I_{2}\\}$ and

$\varepsilon(a_{1},b_{1})=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\quad\forall(a_{1},a_{2})\in
V_{1}\times V_{2}$
$\rho((a_{1},b_{1})(a_{2},b_{2}))=\rho_{1}(a_{1}a_{2})\wedge\rho_{2}(b_{1}b_{2})$
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\eta_{2}(b_{1},b_{1}b_{2})$

######  Theorem 5.2.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two FIGs with
effective pairs, then
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\diamond\tilde{\mathscr{X}_{2}}$
is a FIG with effective pairs.

###### Proof.

Let
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\diamond\tilde{\mathscr{X}_{2}}$
be the tensor product of FIGs, $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$, with effective pairs. Let
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ be an arbitrary pair in
$\tilde{\mathscr{X}}$.
Then by Definition 5.1,

$\displaystyle\begin{split}\eta((a_{1},b_{1}),(a_{1},&b_{1})(a_{2},b_{2}))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\eta_{2}(b_{1},b_{1}b_{2})\\\
&=\varepsilon_{1}(a_{1})\wedge\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})\wedge\rho_{2}(b_{1}b_{2})\\\
&=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(b_{1})\wedge\rho_{1}(a_{1}a_{2})\wedge\rho_{2}(b_{1}b_{2})\\\
&=\varepsilon(a_{1},b_{1})\wedge\rho((a_{1},b_{1})(a_{2},b_{2}))\end{split}$

This implies that $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is an effective
pair and since $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is an arbitrary
pair in $\tilde{\mathscr{X}}$, every pair in $\tilde{\mathscr{X}}$ is an
effective pair. ∎

######  Remark 5.3.

The tensor product of two CFIGs need not be a CFIG as in Example 5.4.

######  Example 5.4.

Consider Fig. 4. The FIG, $\tilde{\mathscr{X}}$ is the tensor product of FIGs,
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$. It is clear that
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are CFIGs but,
$\tilde{\mathscr{X}}$ is not a CFIG.

######  Theorem 5.5.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs, then
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\diamond\tilde{\mathscr{X}_{2}}$
is a SFIG.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs, and
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}\diamond\tilde{\mathscr{X}_{2}}$
be the tensor product of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$. Let
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ be an arbitrary pair in
$\tilde{\mathscr{X}}$. Corresponding to each pair in $\tilde{\mathscr{X}}$,
there exists one pair in each of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$, i.e., for
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))\in\eta^{*}$ , there exist
$(a_{1},a_{1}a_{2})\in\eta_{1}^{*}$ and $(b_{1},b_{1}b_{2})\in\eta_{2}^{*}$
such that
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\eta_{2}(b_{1},b_{1}b_{2})$.
Therefore corresponding to a path, say $P$ in $\tilde{\mathscr{X}}$ there
exist paths $P_{1}$ and $P_{2}$ in $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively, incidence strengths of each of which
is greater than or equal to the incidence strength of $P$ in
$\tilde{\mathscr{X}}$. Also, corresponding to paths $P_{1}$ in
$\tilde{\mathscr{X}_{1}}$ and $P_{2}$ in $\tilde{\mathscr{X}_{2}}$ there
exists a path $P$ in $\tilde{\mathscr{X}}$. And by Definition 5.1, incidence
strengths of $P_{1},P_{2}$ is greater than or equal to the incidence strength
of $P$ in $\tilde{\mathscr{X}}$. Therefore
$ICONN_{\tilde{\mathscr{X}_{1}}}(a_{1},a_{1}a_{2})\wedge
ICONN_{\tilde{\mathscr{X}_{2}}}(b_{1},b_{1}b_{2})\geq
ICONN_{\tilde{\mathscr{X}}}((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$. Now,
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is strong if

$\displaystyle\eta$
$\displaystyle((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))\geq$ $\displaystyle
ICONN_{\tilde{\mathscr{X}}\setminus((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))}((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2})).$

Since $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are SFIGs, every
pair in $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ is strong.
Hence,

$\displaystyle\begin{split}\eta&((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\eta_{1}(a_{1},a_{1}a_{2})\wedge\eta_{2}(b_{1},b_{1}b_{2})\\\
&=ICONN_{\tilde{\mathscr{X}_{1}}}(a_{1},a_{1}a_{2})\wedge
ICONN_{\tilde{\mathscr{X}_{2}}}(b_{1},b_{1}b_{2})\\\ &\geq
ICONN_{\tilde{\mathscr{X}}}((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))\\\ &\geq
ICONN_{\tilde{\mathscr{X}}\setminus{((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))}}((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))\end{split}$

which implies that $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is strong.
Since $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is arbitrary, every pair in
$\tilde{\mathscr{X}}$ is strong, and therefore $\tilde{\mathscr{X}}$ is a
SFIG. ∎

######  Theorem 5.6.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs
without isolated vertices, and $D_{1}$ and $D_{2}$ are SIDS of
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively. Then
$D_{1}\times V2$ and $V_{1}\times D_{2}$ are SIDS of $\tilde{\mathscr{X}}$ and

$\gamma_{IS}\leq\wedge\\{W(D_{1}\times V_{2}),W(V_{1}\times D_{2})\\}.$

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs
without isolated vertices, and $D_{1}$ and $D_{2}$ are SIDS of
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively.
Consider $D_{1}\times V2$, and if $(u,v)\notin D_{1}\times V2$. then $u\notin
D_{1}$. Since $D_{1}$ is a SIDS of $\tilde{\mathscr{X}_{1}}$, there exists
$u^{\prime}\in D_{1}$ such that $(u,uu^{\prime})$ and
$(u^{\prime},uu^{\prime})$ are strong pairs in $\tilde{\mathscr{X}_{1}}$,
i.e., $u^{\prime}$ dominates $u$ in $\tilde{\mathscr{X}_{1}}$. Also, as
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are SFIGs without
isolated vertices, there exists $v^{\prime}\in V_{2}$ such that $v\neq
v^{\prime}$ and $vv^{\prime}\in E_{2}$, $v$ dominates $v^{\prime}$ in
$\tilde{\mathscr{X}_{2}}$.
Therefore, $(u,v)(u^{\prime},v^{\prime})\in E$ and
$((u,v),(u,v)(u^{\prime},v^{\prime}))$ and
$((u^{\prime},v^{\prime}),(u,v)(u^{\prime},v^{\prime}))$ are strong pairs in
$\tilde{\mathscr{X}}$, i.e., $(u^{\prime},v^{\prime})$ dominates $(u,v)$.
Hence $D_{1}\times V_{2}$ is a SIDS in $\tilde{\mathscr{X}}$. Similarly,
$V_{1}\times D_{2}$ is also a SIDS in $\tilde{\mathscr{X}}$.
Therefore, clearly $\gamma_{IS}\leq\wedge\\{W(D_{1}\times V_{2}),W(V_{1}\times
D_{2})\\}$. ∎

##  6 Strong Incidence Domination in Composition of Fuzzy Incidence Graphs

The section studies the composition of two FIGs. Example 6.3 illustrates that
in general the composition of two SFIG need not be a SFIG. Theorem 6.4 proves
a sufficient condition for the composition of two SFIGs to be a SFIG. Theorem
6.5 obtains a bound for the SIDN in the composition. Proposition 6.7 deals
with the composition of CFIGs.

######  Definition 6.1.

Let
$\tilde{\mathscr{X}_{1}}=(V_{1},E_{1},I_{1},\varepsilon_{1},\rho_{1},\eta_{1})$
and
$\tilde{\mathscr{X}_{2}}=(V_{2},E_{2},I_{2},\varepsilon_{2},\rho_{2},\eta_{2})$
be two FIGs. Then the composition of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ denoted as
$\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ is the FIG,
$\tilde{\mathscr{X}}=(V,E,I,\varepsilon,\rho,\eta)$ such that: $V=V_{1}\times
V_{2}$, $E=\\{(a_{1},b_{1})(a_{2},b_{2})|a_{1}=a_{2},\,b_{1}b_{2}\in
E_{2}\text{ \, or \, }a_{1}a_{2}\in E_{1}\\}$ and
$I=\\{((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))|a_{1}=a_{2},(b_{1},b_{1}b_{2})\in
I_{2},(b_{2},b_{1}b_{2})\in I_{2}\text{ \,or\, }(a_{1},a_{1}a_{2})\in
I_{1},(a_{2},a_{1}a_{2})\in I_{1}\\}$ and

$\varepsilon(a_{1},a_{2})=\varepsilon_{1}(a_{1})\wedge\varepsilon_{2}(a_{2})\quad\forall(a_{1},a_{2})\in
V_{1}\times V_{2}$
$\rho((a_{1},b_{1})(a_{2},b_{2}))=\begin{cases}\varepsilon_{1}(a_{1})\wedge\rho_{2}(b_{1}b_{2})&if\;a_{1}=a_{2},b_{1}b_{2}\in
E_{2}\\\
\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\;b_{1}=b_{2},a_{1}a_{2}\in
E_{1}\\\
\rho_{1}(a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})\wedge\varepsilon_{2}(b_{2})&if\;b_{1}\neq
b_{2},a_{1}a_{2}\in E_{1}\end{cases}$
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))=\begin{cases}\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}b_{2})&if\;a_{1}=a_{2},\\\
&\qquad(b_{1},b_{1}b_{2})\in I_{2}\\\
\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})&if\;b_{1}=b_{2},\\\
&\qquad(a_{1},a_{1}a_{2})\in I_{1}\\\
\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})\\\
\wedge\varepsilon_{2}(b_{2})&if\;b_{1}\neq b_{2},\\\
&\qquad(a_{1},a_{1}a_{2})\in I_{1}\end{cases}$

######  Remark 6.2.

In general the composition of two SFIGs need not be a SFIG. Example 6.3 is the
illustration for the same.

######  Example 6.3.

Consider Fig. 5. Consider the cycle $C:(a,v)(b,v)(a,w)(a,v)$ in
$\tilde{\mathscr{X}}$. The cycle $C$ has only one weakest pair
$((a,w),(a,w)(a,v))$. Therefore by Theorem 3.4, $\tilde{\mathscr{X}}$ is not
SFIG.

A sufficient condition for the composition of two SFIGs to be a SFIG is proved
in Theorem 6.4.

######  Theorem 6.4.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs such
that the maximum of weight of pairs in $\tilde{\mathscr{X}_{1}}$ is less than
or equal to the minimum of weight of pairs in $\tilde{\mathscr{X}_{2}}$. Then
the composition $\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ is a SFIG.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs such
that the maximum of weight of pairs in $\tilde{\mathscr{X}_{1}}$ is less than
or equal to the minimum of weight of pairs in $\tilde{\mathscr{X}_{2}}$. Let
$\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ be the composition of
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$. Consider a pair in
$\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$, say
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$. Then, either $a_{1}a_{2}\in
E_{1}$ or $a_{1}=a_{2},\,b_{1}b_{2}\in E_{2}$. The pair is strong if
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))\geq
ICONN_{\tilde{\mathscr{X}}\setminus((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))}((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$.
There are two cases;
Case 1: $a_{1}a_{2}\in E_{1}$.
Since the maximum of weight of pairs in $\tilde{\mathscr{X}_{1}}$ is less than
or equal to the minimum of weight of pairs in $\tilde{\mathscr{X}_{2}}$, in
this case, weight of the pair $((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is

$\displaystyle\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{1})\wedge\varepsilon_{2}(b_{2})$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{2})$

Now, consider any path from $(a_{1},b_{1})$ to $(a_{2},b_{2})$ that consists
of pairs of the form $((a_{1},b_{n}),(a_{1},b_{n})(a_{2},b_{m}))$,
$b_{n},b_{m}\in V_{2}$. The weight of such a pair is

$\displaystyle\eta((a_{1},b_{n}),(a_{1},b_{n})(a_{2},b_{m}))$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{2})\wedge\varepsilon_{2}(b_{n})\wedge\varepsilon_{2}(b_{m})$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{2})$

Hence the strength of any such path is $\leq\eta_{1}(a_{1},a_{1}a_{2})$.
Now, consider a path from $(a_{1},b_{1})$ to $(a_{2},b_{2})$ that consists of
vertices of the form $(a_{m},b_{n})$, $a_{m}\in V_{1},$ $a_{m}\neq
a_{1},a_{2}$ and $b_{n}\in V_{2}$. Then corresponding to any such path say
$P_{1}:(a_{1},b_{1}),(u_{1},v_{1}),...,(u_{k},v_{k}),(a_{2},b_{2})$ in
$\tilde{\mathscr{X}}$, $u_{1},u_{2},...,u_{k}\in\tilde{\mathscr{X}_{1}}$, and
$v_{1},v_{2},...,v_{k}\in\tilde{\mathscr{X}_{2}}$, there exists a walk
$P_{2}:a_{1},u_{1},...,u_{k},a_{2}$ in $\tilde{\mathscr{X}_{1}}$. This walk
together with $\\{(a_{2},a_{1}a_{2}),a_{1}a_{2},(a_{1},a_{1}a_{2})\\}$
consists of a cycle. Since $\tilde{\mathscr{X}_{1}}$ is a SFIG there exists a
pair of weight less than or equal to $\eta_{1}(a_{1},a_{1}a_{2})$. Hence, the
strength of path
$P_{1}\cup\\{(a_{1},b_{1})(a_{2},b_{2}),((a_{2},b_{2}),(a_{1},b_{1})(a_{2},b_{2}))\\}$
is also less than or equal to $\eta_{1}(a_{1},a_{1}a_{2})$. Therefore the pair
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is strong.
Case 2: $a_{1}=a_{2},\,b_{1}b_{2}\in E_{2}$. There are two sub cases;
Sub case 1:
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{1},b_{2}))=\varepsilon_{1}(a_{1})$. Now,
consider a path from $(a_{1},b_{1})$ to $(a_{1},b_{2})$ that consists of
vertices of the form $(a_{1},b_{n})$, $b_{n}\in V_{2}$. Let one such path be
$P_{2}:(a_{1},b_{1}),(a_{1},v_{1}),(a_{1},v_{2}),...,(a_{1},v_{k}),(a_{1},b_{2})$,
$v_{1},v_{2},...,v_{k}\in V_{2}$. The weight of the pairs in the path $P_{2}$
are

$\begin{array}[]{lcl}\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{1},v_{1}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{1},b_{1}v_{1})\\\
\eta((a_{1},v_{1}),(a_{1},b_{1})(a_{1},v_{1}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{1},b_{1}v_{1})\\\
\eta((a_{1},v_{1}),(a_{1},v_{1})(a_{1},v_{2}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{1},v_{1}v_{2})\\\
\eta((a_{1},v_{2}),(a_{1},v_{1})(a_{1},v_{2}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{2},v_{1}v_{2})\\\
\vdots\\\
\eta((a_{1},v_{k}),(a_{1},v_{k})(a_{1},b_{2}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(v_{k},v_{k}b_{2})\\\
\eta((a_{1},b_{2}),(a_{1},v_{k})(a_{1},b_{2}))&=&\varepsilon_{1}(a_{1})\wedge\eta_{2}(b_{2},v_{k}b_{2})\\\
\end{array}$ (1)

Hence the strength of the path is less than or equal to
$\varepsilon_{1}(a_{1})$.
Now, consider a path that consists of vertices of the form $(a_{m},b_{n})$,
$b_{n}\in V_{2}$ and $a_{m}\neq a_{1}$. Then in any such path there exists a
pair of the form $((a_{1},b_{n}),(a_{1},b_{n})(a_{m},v_{k}))$, $v_{k}\in
V_{2}$ and the weight of the pair is

$\displaystyle\eta((a_{1},b_{n}),(a_{1},b_{n})(a_{m},v_{k}))$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{m})\wedge\varepsilon_{2}(b_{n})\wedge\varepsilon_{2}(v_{k})$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{m})$
$\displaystyle\leq\varepsilon_{1}(a_{1})$

Hence, the strength of each such path is less than or equal to
$\varepsilon_{1}(a_{1})$. Therefore, the pair
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is strong.
Sub case 2:
$\eta((a_{1},b_{1}),(a_{1},b_{1})(a_{1},b_{2}))=\eta_{2}(b_{1},b_{1}b_{2})$.
As sub case 1, consider a path say,
$P_{1}:(a_{1},b_{1}),(a_{1},v_{1}),(a_{1},v_{2}),...,(a_{1},v_{k}),(a_{1},b_{2})$,
$v_{1},v_{2},...,v_{k}\in V_{2}$ that consists of vertices of the form
$(a_{1},b_{n})$, $b_{n}\in V_{2}$. Then corresponding to each such path there
exists a path $P_{2}:b_{1},v_{1},v_{2},...,v_{k},b_{2}$ in
$\tilde{\mathscr{X}_{2}}$. Then $P_{2}$ along with
$\\{(b_{2},b_{1}b_{2}),b_{1}b_{2},(b_{1},b_{1}b_{2})\\}$ forms a cycle. Since
$\tilde{\mathscr{X}_{2}}$ is SFIG, there exists a pair of weight
$\leq\eta_{2}(b_{1},b_{1}b_{2})$. Hence by (1), the path
$P_{1}\cup\\{(a_{1},b_{1})(a_{2},b_{2}),((a_{2},b_{2}),(a_{1},b_{1})(a_{2},b_{2}))\\}$
has strength less than or equal to $\eta_{2}(b_{1},b_{1}b_{2})$. Now, consider
a path that consists of vertices of the form $(a_{m},b_{n})$, $b_{n}\in V_{2}$
and $a_{m}\neq a_{1}$. Then in each such path there exists a pair of the form
$((a_{1},b_{n}),(a_{1},b_{n})(a_{m},v_{k}))$, $v_{k}\in V_{2}$ and since the
maximum of weight of pairs in $\tilde{\mathscr{X}_{1}}$ is less than or equal
to the minimum of weight of pairs in $\tilde{\mathscr{X}_{2}}$,

$\displaystyle\eta((a_{1},b_{n}),(a_{1},b_{n})(a_{m},v_{k}))$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{m})\wedge\varepsilon_{2}(b_{n})\wedge\varepsilon_{2}(v_{k})$
$\displaystyle=\eta_{1}(a_{1},a_{1}a_{m})$
$\displaystyle\leq\eta_{2}(b_{1},b_{1}b_{2}).$

Hence every path from $(a_{1},b_{1})$ to $(a_{2},b_{2})$ has strength
$\leq\eta_{2}(b_{1},b_{1}b_{2})$. Therefore, the pair
$((a_{1},b_{1}),(a_{1},b_{1})(a_{2},b_{2}))$ is strong. ∎

######  Theorem 6.5.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be FIGs such that
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ is
SFIG. Let $D_{1}$ and $D_{2}$ be SIDSs of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively. Then $D_{1}\times D_{2}$ is a SIDS of
$\tilde{\mathscr{X}}$ and $\gamma_{IS}\leq W(D_{1}\times D_{2})$.

###### Proof.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be FIGs such that
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ is a
SFIG. Let $D_{1}$ and $D_{2}$ be SIDS of $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ respectively. Consider $(u,v)\notin D_{1}\times
D_{2}$. Then there are three cases;
Case 1: $u\notin D_{1}$ and $v\in D_{2}$.
Since $D_{1}$ is a SIDS of $\tilde{\mathscr{X}_{1}}$, there exists a
$u^{\prime}\in D_{1}$ that dominates $u$. By Definition 6.1,
$(u^{\prime},v)(u,v)\in E$. Since $\tilde{\mathscr{X}}$ is SFIG,
$(u^{\prime},v)$ dominates $(u,v)$.
Case 2: $u\in D_{1}$ and $v\notin D_{2}$.
Since $D_{2}$ is a SIDS of $\tilde{\mathscr{X}_{2}}$, there exists a
$v^{\prime}\in D_{2}$ that dominates $v$. By Definition 6.1,
$(u,v)(u,v^{\prime})\in E$. Since $\tilde{\mathscr{X}}$ is SFIG,
$(u,v^{\prime})$ dominates $(u,v)$.
Case 3: $u\notin D_{1}$ and $v\notin D_{2}$.
Since $D_{1}$ is a SIDS of $\tilde{\mathscr{X}_{1}}$, there exists a
$u^{\prime}\in D_{1}$ that dominates $u$. By Definition 6.1,
$(u^{\prime},v)(u,v)\in E$. Since $\tilde{\mathscr{X}}$ is SFIG,
$(u^{\prime},v)$ dominates $(u,v)$.
Therefore, $D_{1}\times D_{2}$ is a SIDS of $\tilde{\mathscr{X}}$ and hence,
$\gamma_{IS}\leq W(D_{1}\times D_{2})$ ∎

######  Corollary 6.6.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two SFIGs such
that the maximum of weight of pairs in $\tilde{\mathscr{X}_{1}}$ is less than
or equal to the minimum of weight of pairs in $\tilde{\mathscr{X}_{2}}$. Then
$D_{1}\times D_{2}$ is a SIDS of $\tilde{\mathscr{X}}$ and $\gamma_{IS}\leq
W(D_{1}\times D_{2})$, where $D_{1}$ and $D_{2}$ are SIDSs of
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ respectively.

######  Proposition 6.7.

Let $\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ be two CFIGs, then
the composition
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$ is a
CFIG.

###### Proof.

Consider
$\tilde{\mathscr{X}}=\tilde{\mathscr{X}_{1}}[\tilde{\mathscr{X}_{2}}]$, where
$\tilde{\mathscr{X}_{1}}$ and $\tilde{\mathscr{X}_{2}}$ are two CFIGs.
Consider any two distinct vertices $(a,u)$ and $(b,v)$ in
$\tilde{\mathscr{X}}$. There are two cases;
Case 1: $a=b$, Then $u\neq v$. Since $\tilde{\mathscr{X}_{2}}$ is CFIG, it
implies that $uv\in E_{2}$. Therefore by Definition 6.1, $(a,u)(b,v)\in E$.
Case 2: $a\neq b$, In this case, since $\tilde{\mathscr{X}_{1}}$ is CFIG,
$ab\in E_{1}$. Therefore by Definition 6.1, $(a,u)(b,v)\in E$.
Hence by Case 1 and 2, the underlying graph of $\tilde{\mathscr{X}}$ is
complete. Now, consider the edge $(a,u)(b,v)$ in $\tilde{\mathscr{X}}$. Then
weight of $(a,u)(b,v)$ is;

$\displaystyle\rho((a,u)(b,v))$
$\displaystyle=\begin{cases}\varepsilon_{1}(a)\wedge\rho_{2}(uv)\quad&if\quad
a=b\\\ \rho_{1}(ab)\wedge\varepsilon_{2}(u)\quad&if\quad a\neq b\quad and\quad
u=v\\\
\rho_{1}(ab)\wedge\varepsilon_{2}(u)\wedge\varepsilon_{2}(v)\quad&if\quad
a\neq b\quad and\quad u\neq v\end{cases}$
$\displaystyle=\begin{cases}\varepsilon_{1}(a)\wedge\varepsilon_{2}(u)\wedge\varepsilon_{2}(v)\quad&if\quad
a=b\\\
\varepsilon_{1}(a)\wedge\varepsilon_{1}(b)\wedge\varepsilon_{2}(u)\quad&if\quad
a\neq b\quad and\quad u=v\\\
\varepsilon_{1}(a)\wedge\varepsilon_{1}(b)\wedge\varepsilon_{2}(u)\wedge\varepsilon_{2}(v)\quad&if\quad
a\neq b\quad and\quad u\neq v\end{cases}$
$\displaystyle=\varepsilon(a,u)\wedge\varepsilon(b,v)$

Similarly, by Definition 6.1 the weight of the pair $((a,u),(a,u)(b,v))$ is;

$\displaystyle\eta((a,u),(a,u)(b,v))$
$\displaystyle=\begin{cases}\varepsilon_{1}(a)\wedge\eta_{2}(u,uv)\quad&if\quad
a=b\\\ \eta_{1}(a,ab)\wedge\varepsilon_{2}(u)\quad&if\quad a\neq b\quad
and\quad u=v\\\
\eta_{1}(a,ab)\wedge\varepsilon_{2}(u)\wedge\varepsilon_{2}(v)\quad&if\quad
a\neq b\quad and\quad u\neq v\end{cases}$
$\displaystyle=\begin{cases}\varepsilon_{1}(a)\wedge\varepsilon_{2}(u)\wedge\rho_{2}(uv)\quad&if\quad
a=b\\\
\varepsilon_{1}(a)\wedge\rho_{1}(ab)\wedge\varepsilon_{2}(u)\quad&if\quad
a\neq b\quad and\\\ &\qquad u=v\\\
\varepsilon_{1}(a)\wedge\rho_{1}(ab)\wedge\varepsilon_{2}(u)\wedge\varepsilon_{2}(v)\quad&if\quad
a\neq b\quad and\\\ &\qquad u\neq v\end{cases}$
$\displaystyle=\varepsilon(a,u)\wedge\rho((a,u)(b,v))$

By the same way, weight of $((b,v),(a,u)(b,v))$ is
$\eta((b,v),(a,u)(b,v))=\varepsilon(b,v)\wedge\rho((b,v)(a,u))$. Hence,
$\tilde{\mathscr{X}}$ is a CFIG. ∎

##  7 Application

In the research world it is very common for research groups to collaborate and
discuss research interests within the groups. Here, as an application of
tensor product, a collaboration graph is considered.
Consider two research groups. Each group is represented by a FIG with vertices
representing the people in the group. An edge joins two vertices $a$ and $b$
if person $a$ is involved in a research discussion with person $b$. Each
vertex is assigned with a weight 1, representing their involvement in the
discussion. The edge $ab$ represent the discussion between $a$ and $b$ and
edge weight $\rho(ab)$ is the total discussions between person $a$ and $b$.
The pair weight represents their individual contribution to the discussion
i.e. a pair $(a,ab)$ is the contribution of $a$ to the discussion between $a$
and $b$ . Now, suppose that the two groups wish to collaborate. Now a new
group is formed with vertex representing a team of two people. Vertex in the
new group is of the form $(a,u)$, where person $a$ is from the first group and
person $u$ is from the second group. Vertices $(a,u)$ and $(b,v)$ are adjacent
if $a$ and $b$ are engaged in a discussion in the first group, and $u$ and $v$
are engaged in a discussion in the second group. Now consider the tensor
product of the two graphs. The edge weight is the minimum discussion between
the two teams, and the pair weight is the minimum contribution of each team to
the discussion.
Now, consider Fig. 6. FIGs $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$ represents two research groups. Consider the FIG,
$\tilde{\mathscr{X}}$ with collaboration graph as underlying graph. Each
vertex represent two people, one from each group. Discussion happens between
two teams $(a,u)$ and $(b,v)$ of the group if $a$ and $b$ are involved in a
discussion in the first group and, $u$ and $v$ are involved in a discussion in
the second group. It can be observed that the FIG $\tilde{\mathscr{X}}$ is the
tensor product of FIGs $\tilde{\mathscr{X}_{1}}$ and
$\tilde{\mathscr{X}_{2}}$. Here edge weight $\rho((a,u)(b,v))$ is the minimum
of discussion between $a$ and $b$, and $u$ and $v$. Similarly, the pair weight
$\eta((a,u),(a,u)(b,v))$ is the minimum of individual contribution of $a$ and
$u$ to the discussion $ab$ and $uv$ respectively.

In FIG, $\tilde{\mathscr{X}}$ the concept of SID can also be considered. A
minimum SIDS of $\tilde{\mathscr{X}}$ is $\\{(a,u),(a,v),(b,v),(c,u)\\}$ which
means 4 teams are required for the discussion to happen in the entire group.
And SIDN is 0.6, which is the minimum individual contribution of the 4 teams
in the SIDS to the discussion.

##  8 Conclusion

The introduction of graph operations motivated the researchers to study the
properties of the new graphs obtained from the known structures. Domination is
an area of graph theory of growing importance and is extensively studied by
researchers. Hence it is significant to combine the ideas to study SID in the
operations on FIGs. The article deals with the Cartesian product, join, tensor
product, and composition of FIGs and some of the basic properties of the
obtained graphs. The study mainly focus on SFIGs and FIGs with effective
pairs. SID is also discussed in the FIGs obtained from the operations. Bounds
for the SIDN of FIG are also obtained for each operation.

Acknowledgement

The first author gratefully acknowledges the financial support of Council of
Science and Industrial Research (CSIR), Government of India.
The authors would like to thank the DST, Government of India, for providing
support to carry out this work under the scheme ’FIST’
(No.SR/FST/MS-I/2019/40).

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