# On three domination numbers in block graphs

Aline Parreau  CNRS, LIRIS, University Lyon 1, Lyon, France  Annegret K.
Wagler  LIMOS, University Clermont Auvergne, Clermont-Ferrand, France

###### Abstract

The problems of determining minimum identifying, locating-dominating or open
locating-dominating codes are special search problems that are challenging
both from a theoretical and a computational point of view. Hence, a typical
line of attack for these problems is to determine lower and upper bounds for
minimum codes in special graphs. In this work we study the problem of
determining the cardinality of minimum codes in block graphs (that are
diamond-free chordal graphs). We present for all three codes lower and upper
bounds as well as block graphs where these bounds are attained.

###### keywords:

identifying, locating-dominating and open locating-dominating codes, block
graphs

††thanks: Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
##  1 Introduction

For a graph $G$ that models a facility or a multiprocessor network, detection
devices can be placed at its vertices to locate an intruder (like a fire, a
thief or a saboteur) or a faulty processor. Depending on the features of the
detection devices, different dominating sets can be used to determine the
optimal distribution of the detection devices in $G$. In the following, we
study three problems arising in this context which all have been actively
studied during the last decade, see e.g. the bibliography maintained by
Lobstein [20].

Let $G=(V,E)$ be a graph. The (open) neighborhood of a vertex $u$ is the set
$N(u)$ of all vertices of $G$ adjacent to $u$, and $N[u]=\\{u\\}\cup N(u)$ is
the closed neighborhood of $u$.

A subset $C\subseteq V$ is an _identifying code_ (for short: ID-code) of $G$
if

  * •

$N[u]\cap C\neq\emptyset$ for all $u\in V$ (domination),

  * •

$N[u]\cap C\neq N[v]\cap C$ for all $u,v\in V$ (separation),

see Figure 1(a) for an example. Identifying codes were introduced in [19]. Not
every graph $G$ admits an identifying code, i.e. is _identifiable_ : this
holds if and only if there are no true twins in $G$ (i.e., there is no pair of
distinct vertices $u,v\in V$ with $N[u]=N[v]$) [19]. On the other hand, for
every identifiable graph, its whole vertex set trivially forms an identifying
code. The _identifying code number_ $\gamma^{ID}(G)$ of a graph $G$ is the
minimum cardinality of an identifying code of $G$.

A subset $C\subseteq V$ is a _locating-dominating code_ (for short: LD-code)
if

  * •

$N[u]\cap C\neq\emptyset$ for all $u\in V$ (domination),

  * •

$N(u)\cap C\neq N(v)\cap C$ for all $u,v\in V-C$ (open-separation),

see Figure 1(b) for an example. Locating-dominating codes were introduced in
[22, 23]. By definition, every graph has a locating-dominating code (as its
whole vertex set trivially forms an LD-code). The _locating-dominating number_
$\gamma^{LD}(G)$ of a graph $G$ is the minimum cardinality of a locating-
dominating code of $G$.

A subset $C\subseteq V$ is an _open locating-dominating code_ (for short: OLD-
code) of $G$ if

  * •

$N(u)\cap C\neq\emptyset$ for all $u\in V$ (open-domination),

  * •

$N(u)\cap C\neq N(v)\cap C$ for all $u,v\in V$ (open-separation),

see Figure 1(c) for an example. Open locating-dominating codes were introduced
in [24]. Not every graph $G$ admits an OLD-code: this holds if and only if
there are neither isolated vertices nor false twins in $G$ (i.e., no pair of
distinct vertices $u,v\in V$ with $N(u)=N(v)$) [24]. On the other hand, the
whole vertex set forms an OLD-code of any connected false twin-free graph. The
_open locating-dominating number_ $\gamma^{OLD}(G)$ of a graph $G$ is the
minimum cardinality of an OLD-code of $G$.

Determining $\gamma^{ID}(G)$ or $\gamma^{LD}(G)$ is in general NP-hard [11]
and remains hard for several graph classes where other in general hard
problems are easy to solve, including bipartite graphs [11] and two classes of
chordal graphs, namely split graphs and interval graphs [12, 16]. Determining
$\gamma^{OLD}(G)$ is also in general NP-hard [24] and remains NP-hard for
perfect elimination bipartite graphs and interval graphs [16], and APX-
complete for chordal graphs with maximum degree 4 [21]. Hence, a typical line
of attack for these problems is to determine minimum identifying or (open)
locating-dominating codes of special graphs. Closed formulas for these
parameters have been found so far only for restricted graph families (e.g. for
paths and cycles [9, 23, 24], for stars [17], for complete multipartite graphs
[3, 5] and for some subclasses of split graphs, including thin headless
spiders [4]). In parallel, lower and upper bounds using the number of vertices
and the VC-dimension of graphs have been determined [10, 15]. More details
will be given in Sections 2 and 3. More results on all three problems are
listed in [20].

Figure 1: A block graph where the black vertices form a minimum (a) ID-code,
(b) LD-code, (c) OLD-code.

In this paper, we consider the family of block graphs. A _block graph_ is a
graph in which every maximal 2-connected subgraph (block) is a clique. Block
graphs are precisely the diamond-free chordal graphs [7] resp. those chordal
graphs in which every two maximal cliques have at most one vertex in common
[18]. In particular, any tree is a block graph. Linear-time algorithms to
compute all three domination numbers in block graphs are presented in [1],
thus block graphs are chordal graphs for which the three problems can be
solved in linear time.

In this paper, we complement this result by determining lower and upper bounds
for all three domination numbers in block graphs. We give bounds using both
the number of vertices – as it has been done for several classes of graphs –
but also using the number $n_{Q}(G)$ of maximal cliques of $G$, that is more
pertinent for block graphs. Note that for a connected block graph $G$ that is
not a single vertex, the two numbers $|V(G)|$ and $n_{Q}(G)$ are related:
indeed, in a block graph, the number of vertices is always larger than the
number of maximal cliques such that

$n_{Q}(G)<|V(G)|$

holds (which is not true in general as e.g. complete bipartite graphs
$K_{n,m}$ with $2\leq n<m$ show). There is no relation of the two numbers in
the other direction (since any clique is a block graph with a single maximal
clique), however, if the graph is identifiable, we can prove that $|V(G)|\leq
2n_{Q}(G)-1$.

In [2] it was conjectured that $\gamma^{ID}(G)$ of any block graph $G$ can be
bounded from above by $n_{Q}(G)$; it was observed that this bound is true for
trees and it was verified for some families of block graphs, including thin
headless spiders and critical identifiable block graphs. Here, we prove the
conjecture for all block graphs and address similar questions for LD- and OLD-
codes.

Concerning the relation of the three studied domination numbers, it is
immediate from the definitions that

$\gamma^{LD}(G)\leq\min\\{\gamma^{ID}(G),\gamma^{OLD}(G)\\}$

as every ID-code and every OLD-code satisfy the conditions for locating-
domination. On the other hand, one can prove that both $\gamma^{ID}(G)$ and
$\gamma^{OLD}(G)$ are smaller than $2\gamma^{LD}(G)$.

Note that $\gamma^{ID}(G)$ and $\gamma^{OLD}(G)$ are incomparable in general,
which remains true on block graphs:

  * •

we have from results in [9, 24] that $\gamma^{ID}(G)<\gamma^{OLD}(G)$ holds
for all paths $P_{n}$ with $n\geq 7$, but

  * •

we have $\gamma^{ID}(S_{n})=n+1$ by [5] and we can show that
$\gamma^{OLD}(S_{n})=n$, thus $\gamma^{ID}(G)>\gamma^{OLD}(G)$ holds for all
thin headless spiders $S_{n}$ with $n\geq 3$.

##  2 Upper bounds

###  2.1 Identifying codes

Gravier and Moncel [17] proved that any identifiable graph with at least one
edge satisfies $\gamma^{ID}(G)\leq|V(G)|-1$. Extremal graphs reaching this
bound have been characterized in [14]. In particular, stars, that are trees
and thus block graphs, reach this bound, hence this bound cannot be improved
for block graphs. On the other hand, stars have many maximal cliques, namely
$|V(G)|-1$, so we wonder whether we can improve the bound using $n_{Q}(G)$ as
conjectured in [2]. The answer is affirmative, as proven in the next theorem.

######  Theorem 2.1.

Let $G$ be an identifiable block graph. Then $\gamma^{ID}(G)\leq n_{Q}(G)$.

######  Proof 2.2.

In this proof, we will use the word twin for true twin (i.e. vertices with the
same closed neighborhood). We recall that a graph is identifiable if and only
if it is twin-free. The elements of an identifying code are called code
vertices. Two vertices $u,v$ are separated by a vertex $c$ if $c$ is in the
closed neighborhood of exactly one vertex among $\\{u,v\\}$.

Assume by contradiction that there is a twin-free block graph $G$ with
$\gamma^{ID}(G)>n_{Q}(G)$ and let $G$ be such a counterexample with the
minimum number of vertices. Note that $G$ has at least four vertices since it
can be easily checked that the theorem is true for graphs with three vertices.
First note that $G$ has a vertex $x$ of degree 1 (take any vertex of a leaf-
clique which is not the articulation vertex). Let $y$ be the unique neighbor
of $x$ and let $G^{\prime}=G-x$. Then $G^{\prime}$ is a block graph with
$n_{Q}(G^{\prime})<n_{Q}(G)$ (since $G^{\prime}$ is not a single vertex).

Case 1 : $G^{\prime}$ is twin-free. By minimality of $G$, there is an
identifying code $C^{\prime}$ of $G^{\prime}$ of size at most $n_{Q}(G)-1$. If
$y\notin C^{\prime}$, then $C=C^{\prime}\cup\\{x\\}$ is an identifying code of
$G$. Indeed, $x$ is separated from all the vertices except $y$ by itself and
from $y$ by any vertex dominating $y$ in $C^{\prime}$. Any other pair of
vertices was separated in $C^{\prime}$ and is still separated in $C$. Then we
have a contradiction since $C$ is an identifying code of size at most
$n_{Q}(G)$ of $G$. So we can assume that $y\in C^{\prime}$. If $y$ has a
neighbor $z$ in $C^{\prime}$, then again $C^{\prime}\cup\\{x\\}$ is an
identifying code of $G$. Otherwise, it means that $N[y]\cap
C^{\prime}=\\{y\\}$ and by definition of an identifying code, $y$ is the only
vertex with only $y$ as code vertex in its closed neighborhood. Then
$C^{\prime}\cup\\{z\\}$ where $z$ is any neighbor of $y$ but not $x$ is an
identifying code. Indeed, $x$ is separated from all the other vertices since
it is the only one to have precisely code vertex $y$ in its closed
neighborhood, and all the other pairs of vertices where separated in
$C^{\prime}$ so they are still separated.

Case 2 : $G^{\prime}$ has twins. Let $u$, $v$ be a pair of twins of
$G^{\prime}$. They were not twins in $G$, thus $x$ is adjacent to exactly one
of them, without loss of generality, let $x$ be adjacent to $u$ but not to
$v$. Thus $u=y$. Note that $v$ is then unique since if $u$, $v^{\prime}$ were
twins in $G^{\prime}$ then $v$ and $v^{\prime}$ would be twins in $G$. Then
$G^{\prime\prime}=G^{\prime}-v$ is twin-free since $v$ and $u$ have the same
closed neighborhood in $G^{\prime}$, if one of them separates a pair of
vertices, so does the other. By minimality of $G$, there is an identifying
code $C^{\prime\prime}$ of $G^{\prime\prime}$ of size at most
$n_{Q}(G^{\prime\prime})<n_{Q}(G)$. If $y\notin C^{\prime\prime}$, then
$C^{\prime\prime}\cup\\{x\\}$ is an identifying code of $G$. Indeed $x$ is the
unique vertex having exactly $\\{x\\}$ has identifying set. Thus $y$ and $v$
are separated by $x$. Vertex $y$ is still separated from all the other
vertices and so is $v$ since they have the same closed neighborhood except
$x$. If $y\in C^{\prime\prime}$, then $C=C^{\prime\prime}-y\cup\\{x,v\\}$ is
an identifying code. Indeed, as before, $x$ is the unique vertex having
exactly $\\{x\\}$ has identifying set. Vertices $y$ and $v$ are separated by
$x$. Vertex $y$ is the only vertex having $x$ and $v$ in its closed
neighborhood. Finally $v$ is separated from any vertex $t\neq x,y$ by any
vertex $c$ that was separating $t$ and $y$ in $C^{\prime\prime}$.

In all cases, there is an identifying code of size at most $n_{Q}(G)$ in $G$,
a contradiction.

Note that, besides for stars [17], this bound is attained for e.g. thin
headless spiders [4]. Note further that this bound does not hold for general
graphs and even not for chordal graphs since the chordal graph $P_{2k}^{k-1}$
is identifiable, has only two maximal cliques, but needs $2k-1$ vertices in
any identifying code [14].

###  2.2 (Open) locating dominating codes

There are false twin-free graphs having $\gamma^{OLD}(G)=|V(G)|$ which even
remains true if we restrict our considerations to block graphs, as
$\gamma^{OLD}(P_{2})=2$ and $\gamma^{OLD}(P_{4})=4$ holds. However, we can
show that the only block graphs that reach the upper bound $|V(G)|$ are
$P_{2}$ and $P_{4}$.

######  Theorem 2.3.

Let $G=(V,E)$ be a connected false twin-free block graph different from
$P_{2}$ and $P_{4}$. Then $\gamma^{OLD}(G)\leq|V(G)|-1$.

######  Proof 2.4.

Let $G$ be a connected false twin-free block graph that satisfies
$\gamma^{OLD}(G)=|V(G)|$. Assume that $G$ is neither $P_{2}$ nor $P_{4}$.

Using Bondy’s theorem [6], there is a vertex $x$ such that $G-x$ does not have
false twins. Then $S_{1}=V(G)-\\{x\\}$ is not an OLD-code, because one vertex,
say $y$, is not open-dominated. This means that $y$ has degree $1$ with unique
neighbor $x$.

Now, consider the set $S_{2}=V(G)-\\{y\\}$. Since $G$ is not $P_{2}$, $S_{2}$
is an open-dominating set. But $S_{2}$ is not an OLD-code. It means that there
are two vertices that are not separated by $S_{2}$. Since $G$ has no false
twins, these two vertices are separated by $y$ and necessarily, one of them is
actually the vertex $x$. Let $z$ be the other vertex. We must have
$N(x)=N(z)\cup\\{y\\}$.

Consider now $S_{3}=V(G)-\\{z\\}$. Since $N(z)\subset N(x)$, $S_{3}$ is an
open-dominating set. As before, it is not an OLD-code of $G$ and there are two
vertices in $G$ that are only separated by $z$, and necessarily one of them is
$y$. Let $u$ be the other vertex, it has exactly two neighbors that are $x$
and $z$. Note that the four vertices $x,y,z,u$ induce $P_{4}$. Since $G$ is
not $P_{4}$, there must be at least another vertex $v$ adjacent to these
vertices. Since $y$ and $u$ have degrees $1$ and $2$, $v$ must be adjacent to
$x$ and $z$ (since they have the same neighborhood except $y$). But then there
is a $C_{4}$ induced in $G$, a contradiction since $G$ is a block graph.

Note that a characterization of all graphs satisfying $\gamma^{OLD}(G)=|V(G)|$
can be obtained by extending the argumentation of the previous proof.

There is no hope for a general upper bound on $\gamma^{OLD}(G)$ using uniquely
$n_{Q}(G)$. Indeed, this parameter is well defined for cliques and equal to
$|V(G)|-1$ whereas there is a unique maximal clique.

For locating-dominating codes, from
$\gamma^{LD}(G)\leq\min\\{\gamma^{ID}(G),\gamma^{OLD}(G)\\}$, we conclude that
the upper bound for $\gamma^{ID}(G)$ carries over to $\gamma^{LD}(G)$ whenever
the block graph $G$ is identifiable, but we cannot improve the general upper
bound in the other case:

######  Theorem 2.5.

Let $G=(V,E)$ be a block graph. Then

$\gamma^{LD}(G)\leq\left\\{\begin{array}[]{ll}n_{Q}(G)&\mbox{ if }G\mbox{ is
identifiable,}\\\ |V|-1&\mbox{ otherwise.}\end{array}\right.$

Note that these bounds are attained for stars (which are identifiable) and
cliques (which are not identifiable) by [5].

##  3 Lower bounds

The general lower bound for the size of an identifying code using the number
of vertices is $\gamma^{ID}(G)\geq\lceil\log_{2}(|V(G)|+1)\rceil$ [19].
However, to reach this bound, a graph needs to have a large VC-dimension [10].
Indeed, if a graph has VC-dimension $c$ then any identifying code has size at
least $O(|V(G)|^{1/c})$. The value $1/c$ is not always tight, see for example
the case of line graphs, which have VC-dimension at most $4$ but for which the
tight order for the lower bound is $O(|V(G)|^{1/2})$ [13]. Similar results
hold for LD- and OLD-codes.

Block graphs have VC-dimension at most $2$, and thus, using the previous
result, their (identifying) codes are lower bounded by $O(|V(G)|^{1/2})$. In
the following theorem, we improve this lower bound and give a tight result.

######  Theorem 3.1.

Let $G$ be a block graph. Then we have:

  * •

$\gamma^{ID}(G),\gamma^{OLD}(G)\geq\frac{|V(G)|}{3}+1$,

  * •

$\gamma^{LD}(G)\geq\frac{|V(G)|+1}{3}$.

######  Proof 3.2.

Let $C$ be a minimum code of $G$. Let $C_{1},C_{2},...,C_{k}$ be the connected
components in the subgraph $G[C]$ induced by $C$.

For each vertex $x$, the subset $N(x)\cap C$ (resp. $N[x]\cap C$) is called
its signature if $C$ is a LD-code or an OLD-code (resp. an ID-code).

The vertices of the graph are partitioned into the four following parts:

  * •

$V_{1}=C$ contains the vertices of the code,

  * •

$V_{2}$ contains the vertices that have exactly one vertex of the code in
their neighborhood,

  * •

$V_{3}$ contains the vertices that have neighbors in at least two different
connected components of $C$,

  * •

$V_{4}$ contains the other vertices. Note that all their code neighbors are in
the same connected component and that there are at least two such neighbors.

Let $n_{i}(G)$ denote the number of vertices of degree $i$ in $G$.

######  Claim 1.

We have:

  * •

$|V_{2}|\leq|C|$ if $C$ is an LD-code.

  * •

$|V_{2}|\leq|C|-n_{0}(G[C])$ if $C$ is an ID-code.

  * •

$|V_{2}|\leq|C|-n_{1}(G[C])$ if $C$ is an OLD-code.

By definition of $V_{2}$, each vertex $x\in V_{2}$ has its unique neighbor
$u\in C$ as signature: $N(x)\cap C=\\{u\\}$ if $C$ is an LD-code or an OLD-
code, and $N[x]\cap C=\\{u\\}$ if $C$ is an ID-code. Hence, there can be at
most $|C|$ vertices in $V_{2}$.

Moreover, if $C$ is an ID-code, $u$ cannot be isolated in $G[C]$ (otherwise it
will not be separated from $x$) thus there are at most $|C|-n_{0}(G[C])$
vertices in $V_{2}$.

Finally, if $C$ is an OLD-code, and if $u$ has a neighbor $t$ of degree $1$ in
$G[C]$, then $t$ and $x$ are not separated. Note that $u$ has at most one such
neighbor (if there were two such neighbors $t_{1}$ and $t_{2}$, these two
vertices would not be separated). Thus, there are at most $|C|-n_{1}(G[C])$
vertices in $V_{2}$. $\Diamond$

######  Claim 2.

$|V_{3}|\leq k-1$ where $k$ is the number of connected components of $G[C]$.

Indeed, consider the graph $H$ having as vertex set
$V_{3}\cup\\{u_{1},u_{2},...,u_{k}\\}$ where $u_{1}$, $u_{2}$,… $u_{k}$ are
$k$ new vertices. For each vertex $x$ in $V_{3}$, add an edge between $x$ and
$u_{i}$ if $x$ is adjacent to a vertex of $C_{i}$. If there is a cycle in $H$,
there would be a cycle in $G$ involving two vertices of different connected
components $C_{i}$ and $C_{j}$. By definition of a block graph, this cycle has
to induce a clique, but then $C_{i}$ and $C_{j}$ have to be in the same
connected component in $G[C]$, a contradiction. Thus $H$ is a (bipartite)
tree. Since any vertex in $V_{3}$ connects at least two vertices $u_{i}$ and
$u_{j}$, there are at most $k-1$ such vertices. $\Diamond$

######  Claim 3.

We have in all cases that $|V_{4}|\leq|C|-k$ and in particular

  * •

$|V_{4}|\leq|C|-3k+2n_{0}(G[C])$ if $C$ is an ID-code;

  * •

$|V_{4}|\leq|C|-3k+n_{1}(G[C])$ if $C$ is an OLD-code;

  * •

$|V_{4}|\leq|C|-3k+n_{1}(G[C])+2n_{0}(G[C])$ if $C$ is an LD-code.

Consider a component $C_{i}$ of $G[C]$ and a vertex $x\in V_{4}$ connected to
$C_{i}$ only. Note that $C_{i}$ must have at least two vertices. By definition
of a block graph, $N(x)\cap C_{i}$ is a clique (since $C_{i}$ is connected)
and, moreover, a maximal clique of $C_{i}$ of size at least $2$. Thus, if $x$
and $x^{\prime}$ are two elements of $V_{4}$ connected to $C_{i}$, we must
have $N(x)\cap N(x^{\prime})\cap C_{i}$ of size at most 1 (since the
signatures $N(x)\cap C$ and $N(x^{\prime})\cap C$ (resp. $N[x]\cap C$ and
$N[x^{\prime}]\cap C$) are distinct).

This shows that each vertex $x\in V_{4}$ corresponds to a _unique_ maximal
clique of $G[C]$ of size at least $2$. Since there are at most $|C_{i}|-1$
maximal cliques of size at most $2$ in $G[C_{i}]$, we obtain
$|V_{4}|\leq|C|-k$ in any case (i.e., regardless whether $C$ is an ID-, LD- or
OLD-code).

We now precise the bound further.

Consider a component $C_{i}$ with at least two vertices. We know that there
are at most $|C_{i}|-1$ maximal cliques in $G[C_{i}]$. But if there are no
vertices of degree $1$, then there are less maximal cliques. Indeed, consider
the tree with maximal cliques as vertex set and two maximal cliques are
adjacent if they share one vertex. It follows that the number of maximal
cliques is the number of edges plus one of this tree. The number of edges is
at most the number of vertices that are in at least two cliques. Otherwise
said, $n_{Q}(G[C_{i}])\leq|C_{i}|-|V_{LQ}|+1$ where $V_{LQ}$ denotes the
vertices that are in only one maximal clique. Let $n_{LQ}$ be the number of
leaf cliques of $C_{i}$. Since $C_{i}$ has size $2$, we have $n_{LQ}\geq 2$.
There are $n_{1}(G[C_{i}])$ leaf cliques of size $2$, and each one gives one
vertex in $V_{LQ}$. The others give at least two vertices in $V_{LQ}$.
Finally, $|V_{LQ}|\geq n_{1}(G[C_{i}])+2(n_{LQ}-n_{1}(G[C_{i}]))\geq
4-n_{1}(G[C_{i}])$, and $n_{Q}(G[C_{i}])\leq|C_{i}|-3+n_{1}(G[C_{i}])$.

We further have $|V_{4}|\leq\sum_{|C_{i}|\geq
2}(|C_{i}|-3+n_{1}(G[C_{i}]))=|C|-3k^{\prime}+n_{1}(G[C])-n_{0}(G[C])$ where
$k^{\prime}$ denotes the number of components of size at least $2$ in $C$. We
have $k=k^{\prime}+n_{0}(G[C])$ and thus
$|V_{4}|\leq|C|-3k+n_{1}(G[C])+2n_{0}(G[C])$.

In the case of an OLD-code, there are no isolated vertices in $G[C]$ (they
would not be open-dominated), and thus $n_{0}(G[C])=0$.

In the case of an ID-code, the leaf cliques cannot be the signature of a
vertex in $V_{4}$ and there is exactly one vertex of degree $1$ per leaf
clique (since the graph $G[C]$ is identifiable). Thus one can remove the term
$n_{1}(G[C])$. $\Diamond$

In conclusion, we compute the total number of vertices
$n=|C|+|V_{2}|+|V_{3}|+|V_{4}|$ and discuss when the maximum is attained.

If $C$ is an ID-code, then

$\begin{array}[]{rcl}n&=&|C|+|V_{2}|+|V_{3}|+|V_{4}|\\\
&\leq&|C|+|C|-n_{0}(G[C])+k-1+|C|-3k+2n_{0}(G[C])\\\
&=&3|C|-2k-1+n_{0}(G[C])\\\ \end{array}$

follows. Since $n_{0}(G[C])\leq k$, the maximum is attained when $k=1$ and
then $n_{0}(G[C])=0$.

If $C$ is an OLD-code, then

$\begin{array}[]{rcl}n&=&|C|+|V_{2}|+|V_{3}|+|V_{4}|\\\
&\leq&|C|+|C|-n_{1}(G[C])+k-1+|C|-3k+n_{1}(G[C])\\\ &=&3|C|-2k-1\\\
\end{array}$

follows. The maximum is reached when $k=1$.

If $C$ is an LD-code, then

$\begin{array}[]{rcl}n&=&|C|+|V_{2}|+|V_{3}|+|V_{4}|\\\
&\leq&|C|+|C|+k-1+|C|-k\\\ &=&3|C|-1\\\ \end{array}$

follows (and the value is independent from the number $k$ of components of
$G[C]$).

Extremal cases where these bounds are attained can be constructed as follows
(see Fig. 2): Consider the graph with one path $u_{1},....,u_{k}$ (the
vertices in the code $C$) and attach further vertices:

  * •

for an ID-code $C$: attach a single vertex to each $u_{i}$ and vertices to the
pairs $u_{i},u_{i+1}$ for $1<i<k-1$,

  * •

for an OLD-code $C$: attach a single vertex to $u_{1}$, $u_{k}$ and each
$u_{i}$ for $2<i<k-1$ and vertices to all the pairs $u_{i},u_{i+1}$,

  * •

for an LD-code $C$: attach a single vertex to each $u_{i}$ and vertices to all
the pairs $u_{i},u_{i+1}$.

Note that the here presented graphs are all extremal cases for ID-codes,
whereas further extremal graphs exist for OLD-codes and for LD-codes. In fact,
the reasoning from the proof of Theorem 3.1 can be used to characterize the
extremal graphs for all three cases.

Figure 2: Extremal cases where the lower bounds are attained, black vertices
form a minimum (a) ID-code, (b) OLD-code, (c) LD-code.

If we now consider the parameter $n_{Q}(G)$, we can use the relation
$|V(G)|\geq n_{Q}(G)+1$ to obtain a similar lower bound. But this lower bound
can be improved:

######  Theorem 3.3.

Let $G$ be a block graph. Then we have:

  * •

$\gamma^{ID}(G),\gamma^{OLD}(G)\geq\frac{3(n_{Q}(G)+2)}{7}$,

  * •

$\gamma^{LD}(G)\geq\frac{n_{Q}(G)+2}{3}$,

  * •

$\gamma^{OLD}(G)\geq\frac{n_{Q}(G)+3}{2}$.

######  Proof 3.4.

The proof uses the notations $V_{1}$, $V_{2}$, $V_{3}$, $V_{4}$, $n_{0}$,
$n_{1}$ from the proof of Theorem 3.1. We will bound the number of maximal
cliques. There are four types of maximal cliques:

  1. (1)

Maximal cliques that are maximal cliques of $C$ (of size at least $2$) or a
maximal clique of $C$ with one vertex of $V_{4}$. There are at most
$n_{Q}(G[C])-n_{0}(G[C])\leq|C|-k$ of them ($|C_{i}|-1$ for each of
component).

  2. (2)

Maximal cliques of size $2$ between $V_{2}$ and its unique neighbor in the
code. There are at most $|V_{2}|$ such cliques.

  3. (3)

Maximal cliques formed with a vertex of $V_{3}$ and some vertices of the code.

  4. (4)

Maximal cliques that are included in $V(G)\setminus C$.

To count the cliques of the two last types, we consider the same auxiliary
graph $H$ as in the proof of Claim 2. Let $\ell$ be the number of components
in this graph that is actually a forest. The number of cliques of type (iii)
is at most the number of edges in this graph that is $|V_{3}|+k-\ell$. The
number of cliques of type (iv) is at most $\ell-1$ since any such clique has
to connect two connected components of the graph obtained from $G[C\cup
V_{3}]$ where edges inside $V_{3}$ are removed. Since each vertex
corresponding to a vertex of $V_{3}$ in $H$ has degree at least $2$ in $H$,
there are at least $2|V_{3}|$ edges in $H$ which implies that $|V_{3}|\leq
k-\ell$. Finally, the number of cliques of type (iii) and (iv) is at most
$2(k-\ell)\leq 2k-2$. This implies

$n_{Q}(G)\leq n_{Q}(G[C])-n_{0}(G[C])+|V_{2}|+2k-2.$

We now discuss this inequality using the different upper bounds for $|V_{2}|$
proved in Claim 1 and using the fact that $n_{Q}(G[C])-n_{0}(G[C])\leq|C|-k$.

For ID-codes, we obtain

$n_{Q}(G)\leq 2|C|+k-n_{0}(G[C])-2$

and this value is maximized when $k-n_{0}(G[C])$ is maximal which happens when
the number of components of size at least $2$ is maximal. In an identifying
code, this component must have size at least $3$, and thus we have
$k-n_{0}(G[C])\leq|C|/3$ and $n_{Q}(G)\leq 7|C|/3-2$.

For LD-codes, we obtain

$n_{Q}(G)\leq 2|C|+k-2$

and this value is maximized when $k$ is maximal, i.e. $k=|C|$ and then
$n_{Q}(G)\leq 3|C|-2$.

For OLD-codes, we obtain, using the bound
$n_{Q}(G[C])-n_{0}(G[C])\leq|C|-3k+n_{1}(G[C])$ proved for OLD-codes in Claim
3, that

$n_{Q}(G)\leq|C|-3k+n_{1}(G[C])+|C|-n_{1}(G[C])+2k-2=2|C|-k-2$

and this value is maximal when $k=1$ and we finally have $n_{Q}(G)\leq
2|C|-3$.

Note that for trees, since we have $n_{Q}(G)=|E(G)|=|V(G)|-1$, these bounds
are equivalent to the known lower bounds using the number of vertices (see [8]
for ID-codes, [22] for LD-codes and [24] for OLD-codes). In particular, there
are infinite families of trees reaching the three bounds.

Moreover, there is no such lower bound for general graphs. Indeed, consider
the following split graph $G$ with vertex set
$V=\\{v_{1},...,v_{k}\\}\cup\\{u_{X},X\subseteq\\{1,...,k\\},X\neq\emptyset\\}$.
Vertices $v_{1},...v_{k}$ induce a clique whereas vertices $u_{X}$ induce an
independent set. Moreover, there is an edge between $v_{i}$ and $u_{X}$ iff
$i\in X$. This graph has an identifying code of size $2k$ (the clique with the
vertices corresponding to the singletons) but the number of maximum cliques is
$2^{k}$.

##  4 Concluding remarks

The three here studied domination problems are challenging both from a
theoretical and a computational point of view and even remain hard for several
graph classes where other in general hard problems are easy to solve,
including bipartite graphs and chordal graphs. Block graphs form a subclass of
chordal graphs for which all three domination problems can be solved in linear
time [1]. In this paper, we complement this result by presenting for all three
codes lower and upper bounds. We give bounds using both the number of vertices
– as it has been done for several classes of graphs – but also using the
number $n_{Q}(G)$ of maximal cliques of $G$, that is more pertinent for block
graphs. In particular, we verify a conjecture from [2] concerning an upper
bound for $\gamma^{ID}(G)$. Moreover, we address the questions to find block
graphs where the provided lower and upper bounds are attained.

It is interesting whether similar ideas work for graph classes with a similar
structure, e.g. for cacti (graphs in which every maximal 2-connected subgraph
is an edge or a cycle) or for block-cacti (graphs in which every maximal
2-connected subgraph is a clique or a cycle).

Acknowledgement. We would like to thank Florent Foucaud for interesting
discussions on the topic, in particular concerning lower bounds for the
studied domination numbers.

## References

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        https://www.lri.enst.fr/~lobstein/debutBIBidetlocdom.pdf

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# Photonuclear reactions ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ and
${{}^{63}\rm{Cu}}(\gamma,xn)^{63-x}\rm{Cu}$ cross-sections
in the energy range $E_{\rm{\gamma max}}$ = 35–94 MeV

O.S. Deiev, I.S. Timchenko<EMAIL_ADDRESS>S.M. Olejnik, S.M. Potin,
V.A. Kushnir, V.V. Mytrochenko, S.A. Perezhogin, B.I. Shramenko  National
Science Center ”Kharkov Institute of Physics and Technology”,
1, Akademichna St., 61108, Kharkiv, Ukraine

###### Abstract

The bremsstrahlung flux-averaged cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ for the photonuclear reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ and
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ have been measured in the range of
bremsstrahlung end-point energies $E_{\rm{\gamma max}}$ = 35–94 MeV. The
experiments were performed with the electron beam from the NSC KIPT linear
accelerator LUE-40 with the use of the activation and off-line $\gamma$-ray
spectrometric technique. The calculation of the flux-average cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle_{\rm{th}}$ was carried out using
the cross-section $\sigma(E)$ values from the TALYS1.95 code with the default
options.
It is shown that the experimental average cross-sections for the reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ are systematically higher than the
theoretical estimates in the TALYS1.95 code. The obtained
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ supplement the data of different
laboratories for the case of reactions $(\gamma,n)$ and $(\gamma,2n)$. For the
reaction ${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$, the values of
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ were measured for the first
time.

${{}^{63}\rm{Cu}}$, ${{}^{65}\rm{Cu}}$, photonuclear reactions, bremsstrahlung
flux-averaged cross-section, bremsstrahlung end-point energies of 35–94 MeV,
activation and off-line $\gamma$-ray spectrometric technique, TALYS1.95,
GEANT4.9.2.

##  I Introduction

Most of the data on cross-sections for photonuclear reactions, which are
important for many fields of science and technology, as well as for various
data files (EXFOR, RIPL, ENDF, etc.), were obtained in experiments using
quasi-monoenergetic annihilation photons in the energy range of the giant
dipole resonance (GDR). As a rule, such experiments were carried out with the
direct detection of neutrons from the output channel of the reaction. However,
when measuring the neutron yield, it becomes difficult to separate neutrons
registered from various reactions $(\gamma,n)$, $(\gamma,pn)$, $(\gamma,2n)$,
$(\gamma,p2n)$, $(\gamma,3n)$, etc. This can lead to inaccuracies both in the
values of the GDR cross-sections in the high-energy region and the cross-
sections with a higher reaction threshold.

In [1], attention was drawn to the discrepancy between the data on the partial
cross-sections for the reactions $(\gamma,n)$, $(\gamma,2n)$, $(\gamma,3n)$ of
different laboratories [2, 3] in the study of photo-disintegration of the
${}^{181}\rm{Ta}$ nucleus. It is also shown that the sums of the cross-
sections $(\gamma,n)$, $(\gamma,2n)$, $(\gamma,3n)$ obtained in different
laboratories agree satisfactorily. Varlamov’s works [4, 5, 6] noted systematic
discrepancies between the cross-sections of partial photoneutron reactions
$(\gamma,n)$, $(\gamma,2n)$, $(\gamma,3n)$ and concluded that the experimental
data for many nuclei are not sufficiently reliable due to large systematic
errors of the used method of separating neutrons by multiplicity. Based on
experimental data and postulates of a combined model, in [7, 8] an
experimentally-theoretical method was proposed for evaluating the values of
reaction cross-sections and corrections were made between the partial
photoneutron reactions on several nuclei, for example, see [5, 9].

An experimental study of photonuclear reactions on stable isotopes of copper
was carried out in some works [10, 11, 12, 13, 14, 15, 16, 17] and the values
of the cross-sections for the reactions $(\gamma,n)$ and $(\gamma,2n)$ were
obtained. The discrepancy between the data of different laboratories was
observed both in the shape of the energy dependence of the cross-sections (the
width of the distribution and the position of the maximum) and in the absolute
value at the maximum of the cross-sections. It was shown in [18] that the
Fultz’s data [10] are systematically lower than Varlamov’s ones [4], and it
was pointed out that it is necessary to introduce a multiplying factor of 1.17
to the data [10] on the $(\gamma,n)$ reactions for the ${}^{63}\rm{Cu}$ and
${}^{65}\rm{Cu}$ nuclei. In the review paper [19], regarding [4, 5], the
cross-section maxima for the reactions
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ are given: 79.79 mb and 86.38 mb,
respectively, which differs significantly from results [10]. An analysis of
the experimental cross-sections on copper isotopes for the $(\gamma,2n)$
reaction from [10], carried out in [5] using the experimentally-theoretical
procedure [7, 8], showed the need for a significant correction of these data.
The authors of [7, 8] also believe that it is necessary to introduce
corrections not only in the case of the Cu nuclei, but also for all
photonuclear cross-sections obtained by the method of direct neutron
detection.

Thus, we can conclude that there is a spread in the values of photonuclear
cross-sections for Cu isotopes in the GDR region obtained in different
laboratories. Considering that data on photonuclear reactions on copper
isotopes sometimes are used to monitor flux of bremsstrahlung quanta, for
example, ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ in [14] and
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ in [20], it is necessary to introduce
more certainty into the experimental values of the cross-sections using other
methods.

In this work, using the $\gamma$-activation technique and off-line
$\gamma$-ray spectrometric technique, we study photonuclear reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ in the bremsstrahlung end-point
energy range $E_{\rm{\gamma max}}$ = 35–94 MeV. The obtained experimental
results on the average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ of the reactions will make it possible to supplement the data
of different laboratories for the case of the reactions $(\gamma,n)$ and
$(\gamma,2n)$. The cross-sections for the reaction
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ have not been measured before. The
analysis of the found experimental $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ is carried out using the cross-sections $\sigma(E)$ from the
TALYS1.95 code [21] and the data available in the literature.

##  II Flux-average cross-sections determination

###  II.1 Experimental setup and procedure

Experimental study of copper photodisintegration cross-sections has been
carried out through measurements of the residual $\gamma$-activity of the
irradiated sample, which enabled one to obtain simultaneously the data from
different channels of photonuclear reactions. This technique is well known and
has been described in a variety of papers concerned with the investigation of
multiparticle photonuclear reactions, e.g., on the nuclei
${{}^{27}\\!\rm{Al}}$ [22, 23], ${{}^{93}\rm{Nb}}$ [24, 25, 26],
${{}^{181}\rm{Ta}}$ [1, 27].

The schematic block diagram of the experimental setup is presented in Fig. 1.
The $\gamma$-ray bremsstrahlung beam was generated by means of the NSC KIPT
electron linac LUE-40 RDC “Accelerator” [28, 29]. Electrons of the initial
energy $E_{e}$ were incident on the target-converter made from 1.05 mm thick
natural tantalum plate, measuring 20 by 20 mm. To remove electrons from the
bremsstrahlung flux, a cylindrical aluminum absorber, 100 mm in diameter and
150 mm in length, was used. The targets of diameter 8 mm, placed in the
aluminum capsule, were arranged behind the Al-absorber on the electron beam
axis. For transporting the targets to the place of irradiation and back for
induced activity registration, the pneumatic tube transfer system was used.
After the irradiated targets are delivered to the measuring room, the samples
are extracted from the aluminum capsule and are transferred one by one to the
detector for the measurements.

Figure 1: The schematic block diagram of the experimental setup. The upper
part shows the measuring room, where the exposed target (red color) and the
target-monitor (blue color) are extracted from the capsule and are arranged by
turn before the HPGe detector for induced $\gamma$-activity measurements. The
lower part shows the accelerator LUE-40, Ta-converter, Al-absorber, exposure
reaction chamber.

The induced $\gamma$-activity of the irradiated targets was registered by the
semiconductor HPGe detector Canberra GC-2018 with the resolutions of 0.8 and
1.8 keV (FWHM) for the $\gamma$-quanta energies $E_{\gamma}$ = 122 and 1332
keV, respectively. Its efficiency was 20% relative to the NaI(Tl) detector, 3
inches in diameter and 3 inches in thickness. The absolute registration
efficiency of the HPGe detector was calibrated with a standard set of gamma-
ray radiation sources: 22Na, 60Co, 133Ba, 137Cs, 152Eu, 241Am.

The bremsstrahlung spectra of electrons were calculated by using the
GEANT4.9.2 code [30] with due regard for the real geometry of the experiment,
where consideration was given to spatial and energy distributions of the
electron beam. The program code GEANT4.9.2, PhysList G4LowEnergy allows one to
perform calculations taking properly into account all physical processes for
the case of an amorphous target. Similarly, GEANT4.9.2, PhysList QGSP-BIC-HP
makes it possible to calculate the neutron yield due to photonuclear reactions
from targets of different thicknesses and atomic charges. In addition, the
bremsstrahlung gamma fluxes were monitored by the yield of the
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$ reaction. For this purpose, the
natural molybdenum target-witness, placed close by the target under study, was
simultaneously exposed to radiation.

The natCu and natMo targets were used in the experiment. The isotopic
composition of copper is a mixture of two stable isotopes: ${}^{63}\rm{Cu}$
(isotopic abundance 69.17%) and ${}^{65}\rm{Cu}$ (isotopic abundance 30.83%).
For ${}^{100}\rm{Mo}$ in our calculations, we have used the percentage value
of isotope abundance equal to 9.63% (see ref. [30]). The admixture of other
elements in the targets did not exceed 0.1% by weight.

In the experiment, Cu and Mo samples were exposed to radiation at end-point
bremsstrahlung energies $E_{\rm{\gamma max}}$ ranging from 35 to 94 MeV with
an energy step of $\sim$5 MeV. The masses of the Cu and Mo targets were,
respectively, 22 mg and 60 mg. The time of irradiation $t_{\rm irr}$ was 30
min for each energy $E_{\rm{\gamma max}}$ value; the time of residual
$\gamma$-activity spectrum measurement $t_{\rm meas}$ was from 30 min to 17–60
h.

Table 1:  Nuclear spectroscopic data of the radio-nuclides reactions from [31]

Nuclear reaction |  $E_{\rm{th}}$, MeV | $T_{1/2}$ |  $E_{\gamma}$, keV |  $I_{\gamma}$, %
---|---|---|---|---
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ | 9.91 | 12.700 $\pm$ 0.002 h | 1345.84 | 0.473 $\pm$ 0.010
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ | 10.86 | 9.74 $\pm$ 0.02 min |  | 1172.9
---
875.68
| 0.34
---
0.150 $\pm$ 0.007
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ | 19.74 | 3.333 $\pm$ 0.005 h |  | 1185.23
---
282.956
656.008
| 3.75 $\pm$ 0.07
---
12.2 $\pm$ 0.3
10.77 $\pm$ 0.18
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ | 31.44 | 23.7 $\pm$ 0.4 min | 1332.5 | 88
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$ | 8.29 |  $65.94\pm 0.01$ h | 739.50 | $12.13\pm 0.12$

The error of intensity $I_{\gamma}$ for the 1172.9 keV $\gamma$-line was
determined as half-value spreads according to the databases of [31] and [32].
In the case of the 1332.5 keV $\gamma$-line, the $I_{\gamma}$-error is absent
in [31, 32], and therefore it was taken to be 0.5%.

The yield and bremsstrahlung flux-averaged cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ of the
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$,
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ reactions were obtained. Table 1
lists the nuclear spectroscopic data of the radionuclide’s reactions according
to the data from [31]: $E_{\rm{th}}$ denotes reaction thresholds; $T_{1/2}$ –
the half-life period of the nuclei-products, $E_{\gamma}$ are the energies of
the $\gamma$-lines under study and their intensities $I_{\gamma}$.

Measuring the yield of photoneutron reactions on copper isotopes immediately
after the end of irradiation is hampered by the high intensity of the 511 keV
$\gamma$-line from positron annihilation. This can lead to random coincidences
in the detection system, which can distort the result of target activity
measurements. The presence of the positron line reduces the convenience of
measurements, which should be attributed to the disadvantages of using the Cu
nuclei as a monitor target.

In this regard, it is necessary either to significantly (up to 400 mm)
increase the distance between the detector and the irradiated target or to
wait several hours for the intense of short-lived residual radiation lines to
weaken. We used both variants of measurements.

To process the spectra and estimate the number of counts of $\gamma$-quanta in
the full absorption peak $\triangle A$, we used the program InterSpec v.1.0.9
[33]. Figure 2 shows the typical gamma-spectrum from reaction products of the
copper target in the $E_{\gamma}$ range from 800 to 1500 keV.

Figure 2: Fragment of the gamma-ray spectrum from the natCu target measured
for the following parameters: $t_{\rm meas}$ = 63218 s, $t_{\rm cool}$ = 19015
s, $E_{\rm{\gamma max}}$ = 94 MeV, $m$ = 21.815 mg. The spectrum fragment
ranges from 800 to 1500 keV. The background $\gamma$-lines peaks are indicated
by the letters $BG$.

The bremsstrahlung gamma-flux monitoring against the
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$ reaction yield was performed by
comparing the experimentally obtained average cross-section values with the
computation data. To determine the experimental $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ values we have used the $\triangle A$ activity value for the
$E_{\gamma}$ = 739.50 keV $\gamma$-line and the absolute intensity
$I_{\rm\gamma}$ = 12.13% (see Table 1). The theoretical values of the average
cross-section $\langle{\sigma(E_{\rm{\gamma max}})}\rangle_{\rm{th}}$ were
calculated using the cross-sections $\sigma(E)$ from the TALYS1.95 code with
the default options. The normalization (monitoring) factor $k_{\rm{monitor}}$,
derived from the ratios of $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle_{\rm{th}}$ to $\langle{\sigma(E_{\rm{\gamma max}})}\rangle$,
represent the deviation of the GEANT4.9.2-computed bremsstrahlung
$\gamma$-flux from the real flux falling on the target. The determined
$k_{\rm{monitor}}$ values were used for normalizing the cross-sections for
other photonuclear reactions. The monitoring procedure has been detailed in
[24, 25].

The Ta-converter and Al-absorber, used in the experiment, generate the
neutrons that can cause the reaction
${}^{100}\rm{Mo}(n,2n)^{99}\rm{Mo}$. Calculations were made of the neutron
energy spectrum and the fraction of neutrons of energies above the threshold
of this reaction, similarly to [34]. The contribution of the
${}^{100}\rm{Mo}(n,2n)^{99}\rm{Mo}$ reaction to the value of the induced
activity of the 99Mo nucleus has been estimated and it has been shown that
this contribution is negligible compared to the contribution of
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$. The contribution of the reaction
${}^{100}\rm{Mo}(\gamma,p)^{99}\rm{Nb}$,
${}^{99}\rm{Nb}\xrightarrow{\beta^{-}}$${}^{99}\rm{Mo}$ is also negligible.

###  II.2 Experimental accuracy of flux-average cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$

The uncertainty in measurements of experimental values of the average cross-
sections $\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ was determined as a
quadratic sum of statistical and systematical errors. The statistical error in
the observed $\gamma$-activity is mainly due to statistics in the total
absorption peak of the corresponding $\gamma$-line, which varies within 1 to
10%. This error varies depending on the $\gamma$-line intensity and the
background conditions of spectrum measurements. The observed activity
$\triangle A$ of the investigated $\gamma$-line depends on the detection
efficiency, the half-life period, and the absolute intensity $I_{\rm\gamma}$.
The background is generally governed by the contribution of the Compton
scattering of quanta.

The systematic errors are associated with the uncertainties of the: 1.
irradiation time $\sim$0.5%; 2. the electron current $\sim$0.5%; 3. the
detection efficiency of the detector 2–3%, which is mainly associated with the
error of the reference sources of $\gamma$-radiation and the choice of the
approximation curve; 4. the half-life $T_{1/2}$ of the reaction products and
the absolute intensity $I_{\rm\gamma}$ of the analyzed $\gamma$-quanta, which
is noted in Table 1; 5. normalization of the experimental data to the
${}^{100}\rm{Mo}(\gamma,n)^{99}\rm{Mo}$ monitor reaction yield up to 5%; 6.
error in calculating the contribution from competing$\gamma$-lines (described
in the text).

Thus, the statistical and systematical errors are the variables, which differ
for different ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ and
${{}^{63}\rm{Cu}}(\gamma,xn)^{63-x}\rm{Cu}$ reactions. The total uncertainty
of the experimental data is given in figures with experimental results.

###  II.3 Calculation formulas for flux-average cross-sections

The cross-sections $\sigma(E)$, averaged over the bremsstrahlung $\gamma$-flux
$W(E,E_{\rm{\gamma max}})$ from the threshold $E_{\rm{th}}$ of the reaction
under study to the end-point energy of the spectrum $E_{\rm{\gamma max}}$,
were calculated with the use of the theoretical cross-section values computed
with the TALYS1.95 code [21], installed on the Ubuntu20.04. The bremsstrahlung
flux-averaged cross-section $\langle{\sigma(E_{\rm{\gamma max}})}\rangle_{\rm
th}$ in a given energy interval was calculated by the formula:

$\langle{\sigma(E_{\rm{\gamma max}})}\rangle_{\rm
th}=\frac{\int\limits_{E_{\rm{th}}}^{E_{\rm{\gamma max}}}\sigma(E)\cdot
W(E,E_{\rm{\gamma max}})dE}{\int\limits_{E_{\rm{th}}}^{E_{\rm{\gamma
max}}}W(E,E_{\rm{\gamma max}})dE}.$ (1)

These theoretical average cross-sections were compared with the experimental
values calculated by the formula:

$\begin{split}\langle{\sigma(E_{\rm{\gamma max}})}\rangle=\\\
\frac{\lambda\triangle A{\rm{\Phi}}^{-1}(E_{\rm{\gamma
max}})}{N_{x}I_{\gamma}\ \varepsilon(1-\exp(-\lambda
t_{\rm{irr}}))\exp(-\lambda t_{\rm{cool}})(1-\exp(-\lambda
t_{\rm{meas}}))},\end{split}$ (2)

where $\triangle A$ is the number of counts of $\gamma$-quanta in the full
absorption peak (for the $\gamma$-line of the investigated reaction),
$\lambda$ is the decay constant ($\rm{ln}2/\textit{T}_{1/2}$); $N_{x}$ is the
number of target atoms, $I_{\gamma}$ is the absolute intensity of the analyzed
$\gamma$-quanta, $\varepsilon$ is the absolute detection efficiency for the
analyzed photon energy, ${\rm{\Phi}}(E_{\rm{\gamma
max}})={\int\limits_{E_{\rm{th}}}^{E_{\rm{\gamma max}}}W(E,E_{\rm{\gamma
max}})dE}$ is the integrated bremsstrahlung flux in the energy range from the
reaction threshold $E_{\rm{th}}$ up to $E_{\rm{\gamma max}}$, $t_{\rm{irr}}$,
$t_{\rm{cool}}$ and $t_{\rm{meas}}$ are the irradiation time, cooling time and
measurement time, respectively. A more detailed description of all the
calculation procedures necessary for the determination of
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ can be found in [24, 25].

##  III  RESULTS AND DISCUSSIONS

###  III.1  Photonuclear reaction ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$

In the ${{}^{65}\rm{Cu}}(\gamma,n)$ reaction the 64Cu nucleus is formed with
half-life $T_{1/2}$ = 12.700$\pm$0.002 h. The 64Cu nucleus undergoes
$\beta^{-}$-decay (39%) with the formation of the 64Zn nucleus and
$\varepsilon^{+}\beta^{+}$-decay (61%) with the formation of the 64Ni nucleus.
In the second case, a gamma-line emission with $E_{\rm{\gamma}}$ = 1345.84 keV
of low-intensity $I_{\gamma}$ = 0.473% is observed.

An experimental study of the cross-section for the reaction
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ was carried out in several works, for
example, [10] and [4, 19]. The results of these works have significant
differences in the value of the GDR maxima: 75$\pm$7 mb [10], 86.38 mb [19]
(see Fig. 3). These data were matched in the region of the GDR maximum in [4]
by introducing a coefficient of 1.17 into the data of [10].

The calculations of the cross-section $\sigma(E)$ for this reaction were
carried out in the TALYS1.95 code with default parameters. As can be seen from
Fig. 3, the calculated cross-section at the maximum is very close to the
experimental values of $\sigma(E)$ from [10], but significantly lower than
Varlamov’s data [4, 19]. The leading edge of the calculated distribution does
not coincide with the data of the two experiments.

Figure 3: The ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ reaction cross-section.
Experimental data of Fultz [10] (empty circles) and Varlamov [4, 19] (filled
circles), curve – calculation in TALYS1.95 code.

The average cross-sections $\langle{\sigma(E_{\rm{\gamma max}})}\rangle$
calculated using the cross-section from TALYS1.95 and our experimental data
for the reaction ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ are shown in Fig. 4.
It can be seen that all experimental points are located noticeably higher than
the calculated curve ($\sim$15–20%), which is evidence in favor of the data of
[4, 19].

Figure 4: Flux-average cross-section $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ of the reaction ${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$. The
curve is calculated using the TALYS1.95 code. Experimental result: squares –
measurements immediately after irradiation, circles – measurements after
cooling.

###  III.2  Photonuclear reaction ${{}^{65}\rm{Cu}}(\gamma,2n)^{63}\rm{Cu}$

As a result of the ${{}^{65}\rm{Cu}}(\gamma,2n)$ reaction, the stable isotope
63Cu is formed. The induced activity method does not allow us to measure the
yield of this reaction. For completeness of the analysis, Fig. 5 shows the
data from [10] and calculation from TALYS1.95 with default parameters. It can
be seen that there is a noticeable difference between the experimental and
calculated cross-sections. The experiment exceeds the theoretical cross-
sections from the TALYS1.95 code by approximately 1.3 times at energies of
19–24 MeV.

In [5], using the experimental-theoretical approach [7, 8], the reliability of
data on the cross-sections for the reaction
${{}^{65}\rm{Cu}}(\gamma,2n)^{63}\rm{Cu}$ from [10] was verified. It is shown
that the estimated (recalculated) values of the cross-sections are
significantly lower than the experimental ones, and the difference was up to
30% in the energy range $E_{\rm\gamma}$ = 19–25 MeV. This value is very close
to the discrepancy between the TALYS1.95 cross-section and the result [10].

Such a significant change (correction) in the cross-section for the
${{}^{65}\rm{Cu}}(\gamma,2n)^{63}\rm{Cu}$ reaction leads to corrections in the
GDR cross-section for energies above 18 MeV.

Figure 5: The ${{}^{65}\rm{Cu}}(\gamma,2n)^{63}\rm{Cu}$ reaction cross-
section. Curve – calculation in the TALYS1.95 code, points – experimental data
from [10].

###  III.3  Photonuclear reaction ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$

In the ${{}^{63}\rm{Cu}}(\gamma,n)$ reaction the ${}^{62}\rm{Cu}$ nucleus is
formed with half-life $T_{1/2}$ = 9.74 m. The ${}^{62}\rm{Cu}$ nucleus
undergoes $\varepsilon^{+}\beta^{+}$-decay (the ${}^{62}\rm{Ni}$ nucleus is
formed) with the emission of several low-intensity gamma-lines. The line of
$\gamma$-radiation with $E_{\rm\gamma}$ = 1172.9 keV and intensity
$I_{\rm\gamma}$ = 0.34% was used in the work.

An experimental study of this reaction was carried out in several works [10,
11, 12, 13, 14] and [4, 19] in the energy range 10–28 MeV. These data differ
in the height of the GDR maximum: the data of [10] are lower than the results
of other laboratories, as in the case of the
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$ reaction. The disagreement in the GDR
maxima between [10] and [4, 19] was 14%, respectively, 70$\pm$7 mb and 79.8
mb.

There is calculation of cross-section in the TALYS1.95 code with default
parameters shown in Fig. 6. In the region of the maximum, there is good
agreement between the TALYS1.95 calculation and the data of [10]. All other
experimental results [11, 12, 13, 14], including [19], are located above the
theoretical estimate. Note that the calculated distribution widths $\sigma(E)$
are noticeably smaller than any experimental data set.

There are the average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ of the ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ reaction
calculated in the TALYS1.95 code with default parameters and the obtained
experimental data in the energy range $E_{\rm{\gamma max}}$ = 35–94 MeV shown
in Fig. 7. There is an excess of experimental $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ over the calculation for the entire energy range and is to be
15–20%. This result agrees with that observed in Fig. 6 discrepancy between
data [11, 12, 13, 14] and cross-sections from TALYS1.95.

Because natural copper targets were used in the study, the effect of the
${{}^{65}\rm{Cu}}(\gamma,3n)^{62}\rm{Cu}$ reaction on the yield of the
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ reaction was estimated. For this
purpose, the values of the cross-section from the TALYS1.95 code were used to
calculate the yield. It was found that the contribution of
${{}^{65}\rm{Cu}}(\gamma,3n)^{62}\rm{Cu}$ increases with increasing energy
$E_{\rm{\gamma max}}$ from 0.16% at 35 MeV to 1.32% at 94 MeV.

Figure 6: The ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ reaction cross-section.
Calculated cross-sections $\sigma(E)$ from the Talys1.95 code (solid curve)
and experimental data: empty diamonds – [12], empty squares – [11], filled
circles – [10], dashed curve – [14].

Figure 7: Flux-average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ for the reaction ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$.
Curve – calculation using Talys1.95, filled circles – our experimental data.

In the natCu target, under the impact of high-energy bremsstrahlung, isotopes
of cobalt and copper are also formed, which have $\gamma$-radiation with
$E_{\rm{\gamma}}$ value close to that used in the study. Thus, 62Co
($E_{\rm{\gamma}}$ = 1172.9 keV, $I_{\rm{\gamma}}$ = 84%, $T_{1/2}$ = 1.5
min), 62mCo ($E_{\rm{\gamma}}$ = 1172.9 keV, $I_{\rm{\gamma}}$ = 98%,
$T_{1/2}$ = 13.91 min), 60Co ($E_{\rm{\gamma}}$ = 1173.24 keV,
$I_{\rm{\gamma}}$ = 99.974%, $T_{1/2}$ = 5.27 years), 60Cu ($E_{\rm{\gamma}}$
= 1173.24 keV, $I_{\rm{\gamma}}$ = 0.26%, $T_{1/2}$ = 23.7 min).

The contributions of such $\gamma$-radiation were estimated using the cross-
sections from the TALYS1.95 code, taking into account the half-lives, the
intensities of the competing lines, and the experimental conditions
(irradiation, cooling, and measurement times). The calculated total activity
of 62Co and 62mCo nuclei ranged from 0 to 1.72% in the range of studied
$E_{\rm{\gamma max}}$. The contribution of radiation from radionuclides 60Cu
and 60Co is insignificant.

The total amount of corrections ranged from 0.24–3.28% and was taken into
account in the data.

###  III.4  Photonuclear reaction ${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$

In the reaction ${{}^{63}\rm{Cu}}(\gamma,2n)$ the ${}^{61}\rm{Cu}$ nucleus is
formed with half-life $T_{1/2}$ = 3.333 h. The ${}^{61}\rm{Cu}$ nucleus
undergoes $\varepsilon^{+}\beta^{+}$-decay with the formation of the
${}^{61}\rm{Ni}$ nucleus. Three lines with $E_{\rm\gamma}$ = 282.956, 656.008,
and 1185.234 keV and intensity $I_{\rm\gamma}$ of 12.2, 10.77, and 3.75%,
respectively, are used for the study.

The calculations of the cross-sections for the ${{}^{63}\rm{Cu}}(\gamma,2n)$
reaction, performed in the Talys1.95 code with default parameters, are shown
in Fig. 8. All experimental results [10, 13, 15] are significantly higher than
the theoretical estimates: the difference between the data [10] and the
calculated cross-sections reaches 3 times at the cross-section maximum. The
energy dependence of the calculated curve is wider than the experiment shows.

Figure 8: The ${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ reaction cross-
section. Calculated cross-sections $\sigma(E)$ from the Talys1.95 code (curve)
and experimental data: empty squares – [11], empty triangles – [13], filled
diamonds – [15], filled circles – [10].

The experimental average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ for the reaction${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ in
the energy range $E_{\rm{\gamma max}}$ = 35–94 MeV are shown in Fig. 9. The
data found for the three gamma lines agree within the experimental error. To
take into account the contribution of the
${{}^{65}\rm{Cu}}(\gamma,4n)^{61}\rm{Cu}$ reaction to the yield of the
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ reaction, calculations were carried
out similarly to the ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ reaction. The
values of the yield for ${{}^{65}\rm{Cu}}(\gamma,4n)^{61}\rm{Cu}$ was
calculated and it was shown that this contribution increases with energy: for
50 MeV it does not exceed 0.7%, and for 94 MeV it reaches 3.2%. The cross-
section for the ${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ reaction was
corrected (reduced) by the corresponding values at each energy $E_{\rm{\gamma
max}}$.

The comparison the experimental cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ and the calculation with the
Talys1.95 code is shown in Fig. 9. As can be seen from the figure, the
experimental $\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ exceeds the
calculated values over the entire energy range $E_{\rm{\gamma max}}$ and it
averaged $\sim$100%.

In [5], the validity of the data for the
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ reaction from [10] was verified. As
in the case of the
${{}^{65}\rm{Cu}}(\gamma,2n)^{63}\rm{Cu}$ reaction, it was shown that the
estimated (recalculated) data are significantly lower than the experimental
data [10] – approximately by a factor of 1.5 at the energy $E$ = 23 MeV.
However, such a significant change in the data from [10] does not lead to
their agreement with the calculation in Talys1.95, the difference between the
calculation and the recalculated data was about 2 times. This value is very
close to the discrepancy between the calculation and the data found in this
work.

Figure 9: Flux-average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ of the ${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ reaction. The
calculation using Talys1.95 code is denoted by solid curve, our experimental
data: filled diamonds – $E_{\rm{\gamma}}$ = 282.956 keV, empty circles –
656.008 keV, filled circles – 1185.234 keV.

###  III.5  Photonuclear reaction ${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$

In the reaction ${{}^{63}\rm{Cu}}(\gamma,3n)$ the ${}^{60}\rm{Cu}$ nucleus is
formed, the half-life of which is $T_{1/2}$ = 23.7 min. The ${}^{60}\rm{Cu}$
nucleus undergoes $\varepsilon^{+}\beta^{+}$-decay with the formation of the
${}^{60}\rm{Ni}$ nucleus. To study this reaction, a $\gamma$-line with
$E_{\rm\gamma}$ = 1332.5 keV and intensity of $I_{\rm\gamma}$ = 88% is used.

Fig. 10 shows the calculated average cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ for reaction
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ from the Talys1.95 code and the
experimental data measured in this work. As can be seen, at $E_{\rm{\gamma
max}}$ = 50–70 MeV, the experimental values of $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ also show an excess over the calculated ones.

The values of the yield for ${{}^{65}\rm{Cu}}(\gamma,5n)^{60}\rm{Cu}$ was
calculated and it was shown that this contribution increases with energy: for
60 MeV it does not exceed 0.7%, and for 94 MeV it reaches 5.1%. As in the case
of the ${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ reaction, it is necessary to
take into account the contribution of $\gamma$-radiation from competing lines
from ${{}^{60}\rm{Co}}$ ($E_{\rm\gamma}$ = 1332.5 keV, $I_{\rm\gamma}$ =
99.986%, $T_{1/2}$ = 5.27 years) and ${{}^{60m}\rm{Co}}$ ($E_{\rm\gamma}$ =
1332.5 keV, $I_{\rm\gamma}$ = 0.24%, $T_{1/2}$ = 10.47 min). The calculated
total activity of ${{}^{60}\rm{Co}}$ and ${{}^{60m}\rm{Co}}$ nuclei ranged
from 0.1 to 0.15% in the range of studied $E_{\rm{\gamma max}}$.

The value of total corrections increases with energy and amounts to 0.1–5.3%.
The cross-section for the
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ reaction was corrected (reduced) by
the corresponding values at each energy $E_{\rm{\gamma max}}$.

Figure 10: The calculated average cross-sections $\langle{\sigma(E_{\rm{\gamma
max}})}\rangle$ for reaction ${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ from
the Talys1.95 code (curve) and the experimental data measured in this work
(filled circles).

##  IV Conclusions

The total bremsstrahlung flux-averaged cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ for the photonuclear reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ and
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ have been measured in the range of
end-point energies $E_{\rm{\gamma max}}$ = 35–94 MeV. The experiments were
performed with the beam from the NSC KIPT electron linear accelerator LUE-40
with the use of the activation and off-line $\gamma$-ray spectrometric
technique. The calculation of flux-average cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle_{\rm th}$ and yields were carried
out using the cross-section values computed with the TALYS1.95 code with the
default options.

The experimental bremsstrahlung flux-averaged cross-sections
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ for the photonuclear reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$ systematically higher than
theoretical TALYS1.95 estimates. The obtained experimental results on
$\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ supplement the data available in
the literature for the case of the reactions $(\gamma,n)$ and $(\gamma,2n)$
and can be used to analyze the discrepancies in the results of different
laboratories when analyzing possible corrections for reaction cross-sections
obtained by direct neutron detection.

It is shown that the obtained experimental data on the reactions
${{}^{65}\rm{Cu}}(\gamma,n)^{64}\rm{Cu}$,
${{}^{63}\rm{Cu}}(\gamma,n)^{62}\rm{Cu}$ agree satisfactorily with the results
of [19], and in the case of
${{}^{63}\rm{Cu}}(\gamma,2n)^{61}\rm{Cu}$, with the data recalculated in [5]
from work [10].

The data for the $\langle{\sigma(E_{\rm{\gamma max}})}\rangle$ reaction
${{}^{63}\rm{Cu}}(\gamma,3n)^{60}\rm{Cu}$ were measured for the first time.

## Acknowlegment

The authors would like to thank the staff of the linear electron accelerator
LUE-40 NSC KIPT, Kharkiv, Ukraine, for their cooperation in the realization of
the experiment.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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