On the super domination number of graphs

TL;DR

This paper investigates the super domination number in graphs, providing formulas and bounds, and explores specific cases like corona and Cartesian product graphs to deepen understanding of this graph invariant.

Contribution

It introduces closed formulas and tight bounds for the super domination number, including special cases for product graphs, advancing theoretical understanding of this graph parameter.

Findings

Derived closed formulas for super domination number.

Established tight bounds based on graph invariants.

Analyzed super domination in corona and Cartesian product graphs.

Abstract

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article, we obtain closed formulas and tight bounds for the super domination number of $G$ in terms of several invariants of $G$. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered.

Equations

$$N(v)\cap \overline{D}=\{u\}.$$

$$1\le \gamma (G)\le \lceil \frac{n}{2}\rceil \le {\gamma }_{\mathrm{sp}}(G)\le n-1.$$

$$\gamma_{\rm sp}(K_{a_{1},a_{2},\ldots,a_{k}})=\left\{\begin{array}[]{ll}n-1&\mbox{ if at most one value }\ a_{i}\text{ is greater than one.}\\ n-2&\mbox{ otherwise. }\end{array}\right.$$

$${\gamma }_{\mathrm{sp}}(P_n)=\lceil \frac{n}{2}\rceil .$$

$$\displaystyle\gamma_{\rm sp}(C_{n})=\left\{\begin{array}[]{ll}\left\lceil\frac{n}{2}\right\rceil,&n\equiv 0,3\pmod{4};\\ \\ \left\lceil\frac{n+1}{2}\right\rceil,&\text{otherwise}.\end{array}\right.$$

$${\gamma }_{\mathrm{sp}}(G)\ge n-{\alpha }^{\prime }(G).$$

$$M=\{u^{\ast }u\in E(G):u^{\ast }\in D^{\ast }\text{ and }u\in \overline{D}\}$$

$${\gamma }_{\mathrm{sp}}(G)\ge n-\beta (G).$$

$$\alpha (G)+\beta (G)=n.$$

$${\gamma }_{\mathrm{sp}}(G)\ge \alpha (G).$$

$${\gamma }_{\mathrm{sp}}(G)\ge {\gamma }_s(G).$$

$$x\mathcal{R}y⟷N[x]=N[y]\text{mbox}\mathrm{or}N(x)=N(y).$$

$${\gamma }_{\mathrm{sp}}(G)\ge n-t.$$

$${\gamma }_{\mathrm{sp}}(G)\ge n-t+1.$$

$$N(u)\setminus N[S\setminus \{u\}]\ne \varnothing .$$

$${\gamma }_{\mathrm{sp}}(G)\le n-\gamma (G).$$

$${\gamma }_{\mathrm{sp}}(G)\le n-\rho (G).$$

$${\gamma }_{\mathrm{sp}}(G)\le \lfloor \frac{n\Delta }{\Delta +1}\rfloor .$$

$${\gamma }_{\mathrm{sp}}(G)=\alpha (G).$$

$${\gamma }_{\mathrm{sp}}(G)\le n-\rho (L(G)).$$

$${\gamma }_{\mathrm{sp}}(G)\le |V(G)\setminus X|=n-|X|=n-|M|=n-\rho (L(G)),$$

$${\gamma }_{\mathrm{sp}}(G\odot H)=n({\gamma }_{\mathrm{sp}}(H)+1).$$

$${\gamma }_{\mathrm{sp}}(G\odot N_r)=nr.$$

$${\gamma }_{\mathrm{sp}}(G\odot H)\le \left|U\cup \left(\bigcup _{i=1}^n{Y}_i\right)\right|=n+\sum _{i=1}^n|Y_i|=n(1+{\gamma }_{\mathrm{sp}}(H)).$$

$${\gamma }_{\mathrm{sp}}(G\odot H)=|W|=|U|+\sum _{i=1}^n|W_i|\ge n(1+{\gamma }_{\mathrm{sp}}(H)).$$

$$\lambda (G)=\underset{S\in \mathcal{S}(G),S^{\ast }\in \mathcal{P}(S)}{\max }\{|X|:X\subseteq S\text{ and }N(X)\cap (\overline{S}\cup S^{\ast })=\varnothing \}.$$

$$\lceil \frac{n{n}^{\prime }}{2}\rceil \le {\gamma }_{\mathrm{sp}}(G\square H)\le n^{\prime }{\gamma }_{\mathrm{sp}}(G)-\lambda (G)(n^{\prime }-{\gamma }_{\mathrm{sp}}(H)).$$

$$W=V(G\square H)\setminus \left(\left(\overline{S}\times V(H)\right)\cup \left(X\times \overline{S^{\prime }}\right)\right)$$

$$N(x^{\ast },y)=\left(\{x^{\ast }\}\times N(y)\right)\cup \left(N(x^{\ast })\times \{y\}\right),$$

$$N(x,y^{\prime })\cap \overline{W}=N(x,y^{\prime })\cap (X\times \overline{S^{\prime }})=\{x\}\times (N(y^{\prime })\cap \overline{S^{\prime }})=\{(x,y)\}.$$

Full text

Keywords: Super domination number; Domination number; Cartesian product; Corona product.

AMS Subject Classification numbers: 05C69; 05C70 ; 05C76

1 Introduction

The open neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$, we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is dominating in $G$ if every vertex in $\overline{D}$ has at least one neighbour in $D$, i.e., $N(u)\cap D\neq\emptyset$ for every $u\in\overline{D}$. The domination number of $G$, denoted by $\gamma(G)$, is the minimum cardinality among all dominating sets in $G$. A dominating set of cardinality $\gamma(G)$ is called a $\gamma(G)$-set. The reader is referred to the books [9, 10] for details on domination in graphs.

A set $D\subseteq V(G)$ is called a super dominating set of $G$ if for every vertex $u\in\overline{D}$, there exists $v\in D$ such that

[TABLE]

If $u$ and $v$ satisfy (1), then we say that $v$ is a private neighbour of $u$ with respect to $\overline{D}$. The super domination number of $G$, denoted by $\gamma_{\rm sp}(G)$, is the minimum cardinality among all super dominating sets in $G$. A super dominating set of cardinality $\gamma_{\rm sp}(G)$ is called a $\gamma_{\rm sp}(G)$-set. The study of super domination in graphs was introduced in [14]. We recall some results on the extremal values of $\gamma_{\rm sp}(G)$.

Theorem 1**.**

[14]*

Let $G$ be a graph of order $n$. The following assertions hold.*

• •

$\gamma_{\rm sp}(G)=n$* if and only if $G$ is an empty graph.*

• •

$\gamma_{\rm sp}(G)\geq\lceil\frac{n}{2}\rceil$.

• •

$\gamma_{\rm sp}(G)=1$* if and only if $G\cong K_{1}$ or $G\cong K_{2}$.*

It is well known that for any graph $G$ without isolated vertices, $1\leq\gamma(G)\leq\lceil\frac{n}{2}\rceil$. As noticed in [14], from the theorem above we have that for any graph $G$ without isolated vertices,

[TABLE]

Connected graphs with $\gamma_{\rm sp}(G)=\frac{n}{2}$ were characterized in [14], while all graphs with $\gamma_{\rm sp}(G)=n-1$ were characterized in [4].

The following examples were previously shown in [14].

1. (a)

For a complete graph $K_{n}$ with $n\geq 2$, $\gamma_{\rm sp}(K_{n})=n-1$. 2. (b)

For a star graph $K_{1,n-1},$ $\gamma_{\rm sp}(K_{1,n-1})=n-1$. 3. (c)

For a complete bipartite graph $K_{r,t}$ with $\min\{r,t\}\geq 2$, $\gamma_{\rm sp}(K_{r,t})=r+t-2$.

The three cases above can be generalized as follows. Let $K_{a_{1},a_{2},\ldots,a_{k}}$ be the complete $k$-partite graph of order $n=\sum_{i=1}^{k}a_{i}$. If at most one value $a_{i}$ is greater than one, then $\gamma_{\rm sp}(K_{a_{1},a_{2},\ldots,a_{k}})=n-1$ as in such a case $K_{a_{1},a_{2},\ldots,a_{k}}\cong K_{n}$ or $K_{a_{1},a_{2},\ldots,a_{k}}\cong K_{n-a_{i}}+N_{a_{i}}$, where $G+H$ denotes the join of graphs $G$ and $H$. As shown in [4], these cases are included in the family of graphs with super domination number equal to $n-1$. On the other hand, it is not difficult to show that if there are at least two values $a_{i},a_{j}\geq 2$, then $\gamma_{\rm sp}(K_{a_{1},a_{2},\ldots,a_{k}})=n-2$. We leave the details to the reader. In summary, we can state the following.

[TABLE]

The particular case of paths and cycles was studied in [14].

Theorem 2**.**

[14]* For any integer $n\geq 3$,*

[TABLE]

[TABLE]

It was shown in [4] that the problem of computing $\gamma_{\rm sp}(G)$ is NP-hard. This suggests that finding the super domination number for special classes of graphs or obtaining good bounds on this invariant is worthy of investigation. In particular, the super domination number of lexicographic product graphs and joint graphs was studied in [4] and the case of rooted product graphs was studied in [13]. In this article we study the problem of finding exact values or sharp bounds for the super domination number of graphs.

The article is organised as follows. In Section 2, we study the relationships between $\gamma_{\rm sp}(G)$ and several parameters of $G$, including the number of twin equivalence classes, the domination number, the secure domination number, the matching number, the $2$-packing number, the vertex cover number, etc. In Section 3, we obtain a closed formula for the super domination number of any corona graph, while in Section 4 we study the problem of finding the exact values or sharp bounds for the super domination number of Cartesian product graphs and express these in terms of invariants of the factor graphs.

2 Relationship between the super domination number and other parameters of graphs

A matching, also called an independent edge set, on a graph $G$ is a set of edges of $G$ such that no two edges share a vertex in common. A largest matching (commonly known as a maximum matching or maximum independent edge set) exists for every graph. The size of this maximum matching is called the matching number and is denoted by $\alpha^{\prime}(G)$.

Theorem 3**.**

For any graph $G$ of order $n$,

[TABLE]

Proof.

Let $D$ be a $\gamma_{\rm sp}(G)$-set and let $D^{*}\subseteq D$ be a set of cardinality $|D^{*}|=|\overline{D}|$ such that for every $u\in\overline{D}$ there exists $u^{*}\in D^{*}$ such that $N(u^{*})\cap\overline{D}=\{u\}.$ Since

[TABLE]

is a matching, we have that $n-\gamma_{\rm sp}(G)=|\overline{D}|=|M|\leq\alpha^{\prime}(G),$ as required. ∎

As a simple example of a graph where the bound above is achieved we can take $G\cong K_{1}+(\cup^{k}K_{2})$. In this case $\alpha^{\prime}(G)=k$, $n=2k+1$ and $\gamma_{\rm sp}(G)=k+1=n-\alpha^{\prime}(G).$ Another example is $\gamma_{\rm sp}(C_{n})=\lceil\frac{n}{2}\rceil=n-\alpha^{\prime}(C_{n})$ whenever $n\equiv 0,3\pmod{4}.$

A vertex cover of $G$ is a set $X\subset V(G)$ such that each edge of $G$ is incident to at least one vertex of $X$. A minimum vertex cover is a vertex cover of smallest possible cardinality. The vertex cover number $\beta(G)$ is the cardinality of a minimum vertex cover of $G$.

Theorem 4**.**

(König [12], Egerváry [5])* For bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover.*

We use Theorems 3 and 4 to derive the following result.

Theorem 5**.**

For any bipartite graph $G$ of order $n$,

[TABLE]

The bound above is attained, for instance, for any star graph and for any hypercube graph. It is well-known that for the hypercube $Q_{k}$, $\beta(Q_{k})=2^{k-1}$ (see, for instance, [8]) and in Section 4 we will show that $\gamma_{\rm sp}(Q_{k})=2^{k-1}$.

An independent set of $G$ is a set $X\subseteq V(G)$ such that no two vertices in $X$ are adjacent in $G$, and the independence number of $G$, $\alpha(G)$, is the cardinality of a largest independent set of $G$.

The following well-known result, due to Gallai, states the relationship between the independence number and the vertex cover number of a graph.

Theorem 6**.**

(Gallai’s theorem)* For any graph $G$ of order $n$,*

[TABLE]

From Theorems 5 and 6 we deduce the following tight bound.

Theorem 7**.**

For any bipartite graph $G$ of order $n$,

[TABLE]

A set $S\subseteq V(G)$ is said to be a secure dominating set of $G$ if it is a dominating set and for every $v\in\overline{S}$ there exists $u\in N(v)\cap S$ such that $(S\setminus\{u\})\cup\{v\}$ is a dominating set. The secure domination number, denoted by $\gamma_{s}(G)$, is the minimum cardinality among all secure dominating sets. This type of domination was introduced by Cockayne et al. in [3].

Theorem 8**.**

For any graph $G$,

[TABLE]

Proof.

Let $S\subseteq V(G)$ be a $\gamma_{\rm sp}(G)$-set. For each $v\in\overline{S}$ let $v^{*}\in S$ such that $N(v^{*})\cap\overline{S}=\{v\}$. Obviously, $S$ and $(S\setminus\{v^{*}\})\cup\{v\}$ are dominating sets. Therefore, $S$ is a secure dominating set and so $\gamma_{s}(G)\leq|S|=\gamma_{\rm sp}(G)$. ∎

The inequality above is tight. For instance, $\gamma_{s}(K_{1,n-1})=\gamma_{\rm sp}(K_{1,n-1})=n-1.$ Moreover, the equality is also achieved whenever $\gamma_{\rm sp}(G)=\gamma(G)$, since $\gamma(G)\leq\gamma_{s}(G)\leq\gamma_{\rm sp}(G)$. This case will be discussed in Corollary 13.

The closed neighbourhood of a vertex $v$ is defined to be $N[v]=N(v)\cup\{v\}$. We define the twin equivalence relation ${\cal R}$ on $V(G)$ as follows:

[TABLE]

We have three possibilities for each twin equivalence class $U$:

1. (a)

$U$ is a singleton set, or 2. (b)

$|U|>1$ and $N(x)=N(y)$ for any $x,y\in U$, or 3. (c)

$|U|>1$ and $N[x]=N[y]$ for any $x,y\in U$.

We will refer to the types (a) (b) and (c) classes as the singleton, false and true twin equivalence classes, respectively.

Let us see three different examples of non-singleton equivalence classes. An example of a graph where every equivalence class is a true twin equivalence class is $K_{r}+(K_{s}\cup K_{t})$, $r,s,t\geq 2$. In this case, there are three equivalence classes composed of $r,s$ and $t$ true twin vertices, respectively. As an example where every class is composed of false twin vertices, we take the complete bipartite graph $K_{r,s}$, $r,s\geq 2$. Finally, the graph $K_{r}+N_{s}$, $r,s\geq 2$, has two equivalence classes and one of them is composed of $r$ true twin vertices and the other one is composed of $s$ false twin vertices. On the other hand, $K_{1}+(K_{r}\cup N_{s})$, $r,s\geq 2$, is an example where one class is singleton, one class is composed of true twin vertices and the other one is composed of false twin vertices.

The following straightforward lemma will be very useful to prove our next theorem.

Lemma 9**.**

Let $G$ be a graph and $D$ a $\gamma_{\rm sp}(G)$-set. Let $D^{*}\subseteq D$ such that $|D^{*}|=|\overline{D}|$ and for every $u\in\overline{D}$ there exists $u^{*}\in D^{*}$ such that $N(u^{*})\cap\overline{D}=\{u\}.$ If $U\subseteq V(G)$ is a twin equivalence class, then $|U\cap\overline{D}|\leq 1$ and $|U\cap D^{*}|\leq 1$.

Theorem 10**.**

For any graph $G$ of order $n$ having $t$ twin equivalence classes,

[TABLE]

Furthermore, if $G$ is connected and $t\geq 3$, then

[TABLE]

Proof.

Let $\{B_{1},B_{2},\dots,B_{t}\}$ be the set of twin equivalence classes of $G$ and let $D$ be a $\gamma_{\rm sp}(G)$-set. By Lemma 9 for every twin equivalence class we have $|B_{i}\cap D|\geq|B_{i}|-1$, which implies that $\gamma_{\rm sp}(G)=|D|\geq\sum_{i=1}^{t}|B_{i}|-t=n-t$.

Suppose that $\gamma_{\rm sp}(G)=n-t$. In such a case, by Lemma 9 we have that $|B_{i}\cap\overline{D}|=1$ and $|B_{i}\cap D^{*}|=1$ for every twin equivalence class. From now on we assume that $G$ is connected and $t\geq 3$. Hence, there exist three twin equivalence classes $B_{i},B_{j},B_{l}$ such that every vertex in $B_{j}$ is adjacent to every vertex in $B_{i}\cup B_{l}$, and six vertices $a,b\in B_{i}$, $x,y\in B_{j}$ and $u,v\in B_{l}$ such that $a,x,u\in\overline{D}$ and $b,y,v\in D^{*}$. Thus, $N(y)\cap\overline{D}\supseteq\{a,u\}$, which is a contradiction. Therefore, the result follows. ∎

The bound $\gamma_{\rm sp}(G)\geq n-t$ is achieved by complete nontrivial graphs $G\cong K_{n}$, complete bipartite graphs $G\cong K_{r,s}$, where $r,s\geq 2$, and by the disjoin union of these graphs. The bound $\gamma_{\rm sp}(G)\geq n-t+1$ is achieved by the graph $G$ shown in Figure 1, where there are four false twin equivalence classes and a singleton equivalence class. In this case, white-coloured vertices form a $\gamma_{\rm sp}(G)$-set.

The open neighbourhood of a set $X\subseteq V(G)$ is defined to be $N(X)=\cup_{x\in X}N(x)$, while the closed neighbourhood is defined to be $N[X]=X\cup N(X)$. A set $S\subseteq V(G)$ is open irredundant if for every $u\in S$,

[TABLE]

Theorem 11**.**

[1]* If a graph $G$ has no isolated vertices, then $G$ has a minimum dominating set which is open irredundant.*

Theorem 12**.**

If a graph $G$ has no isolated vertices, then

[TABLE]

Proof.

If $G$ has no isolated vertices, by Theorem 11, there exists an open irredundant set $S\subseteq V(G)$ such that $|S|=\gamma(G)$. Since every $u\in S$ satisfies (4), we have that for every $u\in S$, there exits $v\in\overline{S}$ such that $N(v)\cap S=\{u\}$, which implies that $\overline{S}$ is a super dominating set. Therefore, $\gamma_{\rm sp}(G)\leq|\overline{S}|=n-\gamma(G).$ ∎

The bound above is tight. For instance, it is achieved by the graph shown in Figure 2. In Section 3 we will show other examples of graphs where the bound above is achieved.

Since $\gamma_{sp}(G)\geq\frac{n}{2}$, we deduce the following consequence of Theorem 12.

Corollary 13**.**

Let $G$ be a graph of order $n$. If $\gamma(G)=\frac{n}{2}$, then $\gamma_{sp}(G)=\frac{n}{2}$.

The converse of Corollary 13 is not true. For instance, as we will see in Section 4, for the Cartesian product of $K_{n}$ times $K_{2}$ we have $\gamma_{sp}(K_{n}\Box K_{2})=n$, while $\gamma(K_{n}\Box K_{2})=2$.

A set $X\subseteq V(G)$ is called a $2$-packing if $N[u]\cap N[v]=\emptyset$ for every pair of different vertices $u,v\in X$. The $2$-packing number $\rho(G)$ is the cardinality of any largest $2$-packing of $G$. It is well known that for any graph $G$, $\gamma(G)\geq\rho(G)$. Meir and Moon [15] showed in 1975 that $\gamma(T)=\rho(T)$ for any tree $T$. We remark that in general, these $\gamma(T)$-sets and $\rho(T)$-sets are not identical.

Corollary 14**.**

Let $G$ be a graph of order $n$. If $G$ does not have isolated vertices,

[TABLE]

To show that the bound above is tight, we can take the graph shown in Figure 2. In Section 3 we will show other examples.

As shown in [18], the domination number of any graph of maximum degree $\Delta$ is bounded below by $\frac{n}{\Delta+1}$. Therefore, the following result is deduced from Theorem 12.

Corollary 15**.**

For any graph $G$ of order $n$ and maximum degree $\Delta$,

[TABLE]

The bound above is achieved, for instance, for any graph with $\gamma_{\rm sp}(G)=n-1$, as in theses cases $\Delta=n-1$. An example of graph with $\Delta<n-1$ and $\gamma_{\rm sp}(G)=\left\lfloor\frac{n\Delta}{\Delta+1}\right\rfloor$ is the one shown in Figure 2.

By Theorems 5 and 12 we deduce the following result.

Theorem 16**.**

Let $G$ be a bipartite graph. If $\gamma(G)=\beta(G)$, then

[TABLE]

Theorem 33 will provide a family of Cartesian product graphs where $\gamma_{\rm sp}(G)=\alpha(G).$

The line graph $L(G)$ of a simple non-empty graph $G$ is obtained by associating a vertex with each edge of $G$ and connecting two vertices of $L(G)$ with an edge if and only if the corresponding edges of $G$ have a vertex in common.

Theorem 17**.**

For any graph $G$ of order $n$,

[TABLE]

Proof.

Let $M$ be a $2$-packing of $L(G)$ such that $|M|=\rho(L(G))$. Let $X,X^{\prime}\subset V(G)$ be two disjoint sets of cardinality $|M|$ such that $|e\cap X|=1$ and $|e\cap X^{\prime}|=1$ for every $e\in M$, i.e., every edge in $M$ has an endpoint in $X$ and the other one in $X^{\prime}$. Since $M$ is a $2$-packing of $L(G)$, for any $u,w\in X$ we have $N(v)\cap X^{\prime}\cap N(w)=\emptyset$. Thus, $V(G)\setminus X$ is a super dominating set of $G$, as for every $u\in X$ there exists $v\in X^{\prime}$ such that $N(v)\cap X=\{u\}$. Hence,

[TABLE]

as required. ∎

To show that the bound above is tight we can take, for instance, both graphs shown in Figure 3. Notice that in these cases Theorem 17 gives a better result than Corollary 14.

3 Super domination in Corona product graphs

The corona product graph $G_{1}\odot G_{2}$ is defined as the graph obtained from $G_{1}$ and $G_{2}$ by taking one copy of $G_{1}$ and $|V(G_{1})|$ copies of $G_{2}$ and joining by an edge each vertex from the $i^{th}$ copy of $G_{2}$ with the $i^{th}$ vertex of $G_{1}$ [6]. It is readily seen that $\gamma(G_{1}\odot G_{2})=\rho(G_{1}\odot G_{2})=|V(G_{1})|$. Therefore, The bounds obtained in Theorem 12 and Corollary 14 are tight, as for the corona graph $G\cong G_{1}\odot K_{r}$ or $G\cong G_{1}\odot N_{r}$, where $G_{1}$ is an arbitrary graph of order $t$ and $\gamma(G)=\rho(G)=t$.

Theorem 18**.**

For any graph $G$ of order $n$ and any nonempty graph $H,$

[TABLE]

Furthermore, for any integer $r\geq 1$,

[TABLE]

Proof.

We first assume that $H$ is nonempty. Let $r$ be the order of $H$, $U=\{u_{1},\dots,u_{n}\}$ the vertex set of $G$, and $V_{i}$ the vertex set of the copy of $H$ associated to $u_{i}$. Let $Y\subset V(H)$ be a $\gamma_{\rm sp}(H)$-set and $Y_{i}\subset V_{i}$ the set associated to $Y$ in the $i^{th}$ copy of $H$. It is readily seen that $U\cup\left(\cup_{i=1}^{n}Y_{i}\right)$ is a super dominating set of $G\odot H$. Thus,

[TABLE]

Now, let $W$ be a $\gamma_{\rm sp}(G\odot H)$-set. If $u_{i}\not\in W$, for some $i$, then $V_{i}\subseteq W$, which implies that $W^{\prime}=(W\setminus V_{i})\cup Y_{i}\cup\{u_{i}\}$ is a super dominating set of $G\odot H$ and $|W^{\prime}|\leq|W|$, as $|Y_{i}|=\gamma_{\rm sp}(H)\leq r-1$. Thus, from now on we can assume that $W$ is taken in such a way that $U\subset W$. Now, let $W_{i}=W\cap V_{i}$. If there exists $u_{i}\in U$ such that $|W_{i}|<|Y_{i}|$, then set of vertices of $H$ associated to $W_{i}$ is a super dominating set of $H$ and $|W_{i}|<\gamma_{\rm sp}(H)$, which is a contradiction. Hence, $|W_{i}|\geq|Y_{i}|=\gamma_{\rm sp}(H)$ for every $i$, which implies that

[TABLE]

Therefore, the first equality holds.

On the other hand, Theorems 3 and 12 imply that $\gamma_{\rm sp}(G\odot N_{r})=nr$. ∎

An alternative proof for the result above can be derived from a formula obtained in [13] for the super domination number of rooted product graphs. We leave the details to the reader.

4 Super domination in Cartesian product graphs

The Cartesian product of two graphs $G$ and $H$ is the graph $G\Box H$ whose vertex set is $V(G\Box H)=V(G)\times V(H)$ and two vertices $(g,h),(g^{\prime},h^{\prime})\in V(G\Box H)$ are adjacent in $G\Box H$ if and only if either

• •

$g=g^{\prime}$ and $hh^{\prime}\in E(H)$, or

• •

$gg^{\prime}\in E(G)$ and $h=h^{\prime}$.

The Cartesian product is a straightforward and natural construction, and is in many respects the simplest graph product [7, 11]. Hamming graphs, which includes hypercubes, grid graphs and torus graphs are some particular cases of this product. The Hamming graph $H_{k,n}$ is the Cartesian product of $k$ copies of the complete graph $K_{n}$. Hypercube $Q_{n}$ is defined as $H_{n,2}$. Moreover, the grid graph $P_{k}\Box P_{n}$ is the Cartesian product of the paths $P_{k}$ and $P_{n}$, the cylinder graph $C_{k}\Box P_{n}$ is the Cartesian product of the cycle $C_{k}$ and the path $P_{n}$, and the torus graph $C_{k}\Box C_{n}$ is the Cartesian product of the cycles $C_{k}$ and $C_{n}$.

This operation is commutative in the sense that $G\Box H\cong H\Box G$, and is also associative in the sense that $(F\Box G)\Box H\cong F\Box(G\Box H)$. A Cartesian product graph is connected if and only if both of its factors are connected.

This product has been extensively investigated from various perspectives. For instance, the most popular open problem in the area of domination theory is known as Vizing’s conjecture. Vizing [17] suggested that the domination number of the Cartesian product of two graphs is at least as large as the product of domination numbers of its factors. Several researchers have worked on it, for instance, some partial results appears in [2, 7]. For more information on structure and properties of the Cartesian product of graphs we refer the reader to [7, 11].

Before obtaining our first result we need to introduce some additional notation. The set of all $\gamma_{\rm sp}(G)$-sets will be denoted by $\mathcal{S}(G)$. For any $S\in\mathcal{S}(G)$ we define the set $\mathcal{P}(S)$ formed by subsets $S^{*}\subseteq S$ of cardinality $|S^{*}|=|\overline{S}|$ such that for every $u\in\overline{S}$ there exists $u^{*}\in S^{*}$ such that $N(u^{*})\cap\overline{S}=\{u\}.$ With this notation in mind we define the following parameter which will be useful to study the super domination number of Cartesian product graphs.

[TABLE]

With the aim of clarifying what this notation means, we consider the graphs shown in Figure 3. For the graph $G_{1}\cong K_{1}+(K_{2}\cup K_{2}\cup K_{1}\cup K_{1})$ (on the left) we have $\gamma_{\rm sp}(G_{1})=5$, $S_{1}=\{1,3,5,6,7\}\in\mathcal{S}(G_{1})$ and $\mathcal{P}(S_{1})=\{\{1,3\}\}$, while for the graph $G_{2}\cong K_{1}+(K_{2}\cup K_{1})$ (on the right) we have that $\gamma_{\rm sp}(G_{2})=3$, $S_{2}=\{1,3,4\}\in\mathcal{S}(G_{2})$ and $\mathcal{P}(S_{2})=\{\{1\},\{4\}\}$. Notice that $\lambda(G_{1})=2$ and $\lambda(G_{2})=1$.

If $G$ has $n$ vertices and $|N(v)|=n-1$, then $v$ is a universal vertex of $G$. It is readily seen that the following remark holds.

Remark 19**.**

Let $v$ be a universal vertex of a graph $G$ of order $n$ and let $S\in\mathcal{S}(G)$. If $v\in\overline{S}\cup S^{*}$ for some $S^{*}\in\mathcal{P}(S)$, then $\gamma_{\rm sp}(G)=n-1.$

For instance, for the graph $G_{2}\cong K_{1}+(K_{2}\cup K_{1})$ shown in Figure 3 we have $\gamma_{\rm sp}(G_{2})=n-1=3$, $S=\{1,3,4\}\in\mathcal{S}(G_{2})$ and $S^{*}=\{4\}\in\mathcal{P}(S)$.

Theorem 20**.**

For any graphs $G$ and $H$ of order $n\geq 2$ and $n^{\prime}\geq 2$, respectively,

[TABLE]

Proof.

The lower bound is deduced from Theorem 1, so we proceed to deduce the upper bound. Let $S$ be a $\gamma_{\rm sp}(G)$-set, $S^{*}\in\mathcal{P}(S)$ and $X\subseteq S$ such that $|X|=\lambda(G)$ and $N(X)\cap(\overline{S}\cup S^{*})=\emptyset$. We claim that for any $\gamma_{\rm sp}(H)$-set $S^{\prime}$, the set

[TABLE]

is a super dominating set of $G\Box H$. To see this we fix $(x,y)\in\overline{W}$. Notice that $x\in\overline{S}$ or $x\in X$, so that we differentiate these two cases.

Case 1: $x\in\overline{S}$. In this case, there exists $x^{*}\in S^{*}$ such that $N(x^{*})\cap\overline{S}=\{x\}$. Since $\{x^{*}\}\times N(y)\subseteq W$, $(N(x^{*})\times\{y\})\cap\overline{W}=(N(x^{*})\cap\overline{S})\times\{y\}=\{(x,y)\}$ and

[TABLE]

we can conclude that $N(x^{*},y)\cap\overline{W}=\{(x,y)\}.$

Case 2: $x\in X$. In this case $N(x)\cap(\overline{S}\cup S^{*})=\emptyset$ and $y\in\overline{S^{\prime}}$. Since $S^{\prime}$ is a super dominating set of $H$, there exists $y^{\prime}\in S^{\prime}$ such that $N(y^{\prime})\cap\overline{S^{\prime}}=\{y\}$. Also, if there exists $w\in N(x)\cap X$, then $S\setminus\{w\}$ is a super dominating set of $G$, which is a contradiction, so that $X$ is an independent set. Hence,

[TABLE]

Therefore, $W$ is a super dominating set of $G\Box H$, which implies that

[TABLE]

as required. ∎

As a direct consequence of Theorem 20 we derive the following bound.

Corollary 21**.**

For any graphs $G$ and $H$ of order $n\geq 2$ and $n^{\prime}\geq 2$, respectively,

[TABLE]

We will see in Theorem 30 that if $n\geq 3$, then $\gamma_{\rm sp}(K_{n}\Box K_{3})=2n$, which implies that the bound given in Corollary 21 is tight, as $\min\{3(n-1),2n\}=2n$ for $n\geq 3$.

The following result is a direct consequence of Theorem 20 and Corollary 21.

Theorem 22**.**

Let $G$ and $H$ be two graphs of order $n\geq 2$ and $n^{\prime}\geq 2$, respectively. If $\gamma_{\rm sp}(G)=\frac{n}{2}$ or $\gamma_{\rm sp}(H)=\frac{n^{\prime}}{2}$, then

[TABLE]

From the result above we have that for any graph $G$ of order $n\geq 2$,

[TABLE]

Since the hypercube graph $Q_{k}$ is defined as $Q_{k}=Q_{k-1}\Box K_{2}$, for $k\geq 2$, and $Q_{1}=K_{2}$, Corollary 22 leads to the following result.

Remark 23**.**

For any integer $k\geq 1$,

[TABLE]

From Theorems 1 and 2, and Corollary 21 we deduce the following result.

Theorem 24**.**

Let $n\geq 3$ be an integer. The following statements hold for any graph $H$ of order $n^{\prime}\geq 2$.

• •

If $n\equiv 0\pmod{2}$, then $\gamma_{\rm sp}(P_{n}\Box H)=\frac{nn^{\prime}}{2}$.

• •

If $n\equiv 1\pmod{2}$, then $\frac{nn^{\prime}}{2}\leq\gamma_{\rm sp}(P_{n}\Box H)\leq\frac{(n+1)n^{\prime}}{2}$.

• •

If $n\equiv 0\pmod{4}$, then $\gamma_{\rm sp}(C_{n}\Box H)=\frac{nn^{\prime}}{2}$.

• •

If $n\equiv 1,3\pmod{4}$, then $\frac{nn^{\prime}}{2}\leq\gamma_{\rm sp}(C_{n}\Box H)\leq\frac{(n+1)n^{\prime}}{2}$.

• •

If $n\equiv 2\pmod{4}$, then $\frac{nn^{\prime}}{2}\leq\gamma_{\rm sp}(C_{n}\Box H)\leq\frac{(n+2)n^{\prime}}{2}$.

As usual in domination theory, when studying a domination parameter, we can ask if a Vizing-like conjecture can be proved or formulated.

Conjecture 25**.**

(Vizing-like conjecture)* For any graphs $G$ and $H$,*

[TABLE]

The above conjecture holds in the following case, which is a direct consequence of Theorem 22.

Remark 26**.**

Let $G$ and $H$ be two graphs of order $n\geq 2$ and $n^{\prime}\geq 2$, respectively. If $\gamma_{\rm sp}(G)=\frac{n}{2}$ or $\gamma_{\rm sp}(H)=\frac{n^{\prime}}{2}$, then

[TABLE]

In order to deduce another consequence of Theorem 20 we need to state the following lemma.

Lemma 27**.**

Let $I(G)$ be the number of vertices of degree one of a graph $G$, and let $S\in\mathcal{S}(G)$. If there exists a universal vertex $v$ of $G$ such that $v\not\in\overline{S}\cup S^{*}$, for some $S^{*}\in\mathcal{P}(S)$, then $\lambda(G)\geq I(G)$.

Proof.

Let $v$ be a universal vertex of $G$. If $I(G)=0$, then we are done, so that we assume that $I(G)>0$.

We first suppose that $\gamma_{\rm sp}(G)=n-1$. If $G\cong K_{1,n-1}$, then for any $S\in\mathcal{S}(G)$ and $S^{*}\in\mathcal{P}(S)$ the universal vertex of $G$ belongs to $\overline{S}\cup S^{*}$. So we assume that $G\not\cong K_{1,n-1}$. In such a case, for any pair of adjacent vertices $x,y\in V(G)\setminus\{v\}$ we have that $S=V(G)\setminus\{x\}\in\mathcal{S}(G)$ and $S^{*}=\{y\}\in\mathcal{P}(S)$, which implies that for any vertex $x$ of degree one, $N(x)\cap(\overline{S}\cup S^{*})=\emptyset$. Hence, $\lambda(G)\geq I(G)$.

Now, suppose that $\gamma_{\rm sp}(G)\leq n-2$. By Remark 19, for any $S\in\mathcal{S}(G)$ and $S^{*}\in\mathcal{P}(S)$ the universal vertex $v$ does not belong to $\overline{S}\cup S^{*}$, which implies that for any vertex $u$ of degree one, $N(u)\cap(\overline{S}\cup S^{*})=\emptyset$. Thus, $\lambda(G)\geq I(G)$. ∎

By Lemma 27 we can derive a consequence of Theorem 20 in which we replace the parameter $\lambda(G)$ by the number of vertices of degree one in $G$.

Proposition 28**.**

Let $I(G)$ be the number of vertices of degree one of a graph $G$ of order $n$, and let $S\in\mathcal{S}(G)$. If there exists a universal vertex $v$ of $G$ such that $v\not\in\overline{S}\cup S^{*}$, for some $S^{*}\in\mathcal{P}(S)$, then for any graph $H$ of order $n^{\prime}$,

[TABLE]

In order to see that the bound above is tight, we can observe that for $G\cong K_{1}+(K_{2}\cup K_{1})$ and $H\cong K_{n^{\prime}}$, $n^{\prime}\geq 3$, we have $\gamma_{\rm sp}(G\Box H)=3n^{\prime}-1=n^{\prime}\gamma_{\rm sp}(G)-I(G)(n^{\prime}-\gamma_{\rm sp}(H)).$

Notice that, by Remark 19, a particular case of Proposition 28 can be stated as follows.

Corollary 29**.**

Let $I(G)$ be the number of vertices of degree one of a graph $G$ of order $n$ and maximum degree $n-1$. If $\gamma_{\rm sp}(G)\leq n-2$, then for any graph $H$ of order $n^{\prime}$,

[TABLE]

We now provide a tight bound on $\gamma_{\rm sp}(G\Box H)$ in terms of the order of $G$ and $H$.

Theorem 30**.**

For any nonempty graphs $G$ and $H$ of order $n\geq 2$ and $n^{\prime}\geq 2$, respectively,

[TABLE]

Furthermore, for any integers $n\geq 4$ and $n^{\prime}\geq 4$,

[TABLE]

and for any integer $n\geq 3$,

[TABLE]

Proof.

Let $x_{1},x_{2}\in V(G)$ and $y_{1},y_{2}\in V(H)$ such that $x_{2}\in N(x_{1})$ and $y_{2}\in N(y_{1})$. Now, let $X\subseteq V(G\Box H)$ such that

[TABLE]

To check that $X$ is a super dominating set of $G\Box H$ we only need to observe that for any $(x,y_{1})\in\overline{X}$ there exists $(x,y_{2})\in X$ such that $N(x,y_{2})\cap\overline{X}=\{x,y_{1}\}$ and for any $(x_{1},y)\in\overline{X}$ there exists $(x_{2},y)\in X$ such that $N(x_{2},y)\cap\overline{X}=\{x_{1},y\}$. Hence, $\gamma_{\rm sp}(G\Box H)\leq|X|=nn^{\prime}-n-n^{\prime}+4$.

To conclude the proof, it remains to consider the Cartesian product of complete graphs. Let $W$ be a $\gamma_{\rm sp}(K_{n}\Box K_{n^{\prime}})$-set. Notice that if $(x,y)\in\overline{W}$, $(a,b)\in W$ and $N(a,b)\cap\overline{W}=\{(x,y)\}$, then $x=a$ leads to $\overline{W}\cap(V(K_{n})\times\{b\})=\emptyset$ and $y=b$ leads to $\overline{W}\cap(\{a\}\times V(K_{n^{\prime}}))=\emptyset$. Furthermore, if $(x,y),(x^{\prime},y)\in\overline{W}$, then $\overline{W}\cap(\{x,x^{\prime}\}\times V(K_{n^{\prime}}))=\{(x,y),(x^{\prime},y)\}$, as for any $y^{\prime}\in V(K_{n^{\prime}})\setminus\{y\}$ we have that $N(x^{\prime},y)\subseteq N[(x,y)]\cup N[(x^{\prime},y^{\prime})]$ and $N(x,y)\subseteq N[(x^{\prime},y)]\cup N[(x,y^{\prime})]$. Analogously, if $(x,y),(x,y^{\prime})\in\overline{W}$, then $\overline{W}\cap(V(K_{n})\times\{y,y^{\prime}\})=\{(x,y),(x,y^{\prime})\}$. Hence,

[TABLE]

Thus, if $n\geq 4$ and $n^{\prime}\geq 4$, then $\gamma_{\rm sp}(K_{n}\Box K_{n^{\prime}})\geq nn^{\prime}-n-n^{\prime}+4$, which implies that $\gamma_{\rm sp}(K_{n}\Box K_{n^{\prime}})=nn^{\prime}-n-n^{\prime}+4$.

Moreover, if $n\geq 3$, then Equation (5) leads to $\gamma_{\rm sp}(K_{n}\Box K_{3})\geq 2n$. To conclude that $\gamma_{\rm sp}(K_{n}\Box K_{3})=2n$ we only need to observe that for any $y,y^{\prime}\in V(K_{3})$ the set $V(K_{n})\times\{y,y^{\prime}\}$ is a super dominating set of $K_{n}\Box K_{3}$. Therefore, the result follows. ∎

The independence number of any Cartesian product graph is bounded below as follows.

Theorem 31**.**

[16]* For any graphs $G$ and $H$ of order $n$ and $n^{\prime}$, respectively,*

[TABLE]

From Theorems 6, 7 and 31 we deduce the following result.

Theorem 32**.**

For any pair of bipartite graphs $G$ and $H$,

[TABLE]

Theorem 33**.**

For any integers $r\geq r^{\prime}\geq 1$,

[TABLE]

Proof.

By Theorem 32, $\gamma_{\rm sp}(K_{1,r}\Box K_{1,r^{\prime}})\geq rr^{\prime}+1$. Next, we proceed to show that $\gamma_{\rm sp}(K_{1,r}\Box K_{1,r^{\prime}})\leq rr^{\prime}+1$. Let $u_{0}$, $v_{0}$, and $U_{0}=\{u_{1},\dots,u_{r}\}$, $V_{0}=\{v_{1},\dots,v_{r^{\prime}}\}$, be the center and the set of leaves of $K_{1,r}$ and $K_{1,r^{\prime}}$, respectively. Let $X=(U_{0}\times\{v_{0}\})\cup(\cup_{j=1}^{r^{\prime}-1}\{(u_{r},v_{j})\})\cup\{(u_{r-1},v_{r^{\prime}})\}$ and $Y=(\{u_{0}\}\times V_{0})\cup(\cup_{i=1}^{r-1}\{(u_{i},v_{1})\})\cup\{(u_{r},v_{r^{\prime}})\}$. Note that, for each vertex $x\in X\subseteq\overline{Y}$, there exists exactly one vertex $y\in Y$ such that $N(y)\cap X=\{x\}$. So, $V(K_{1,r}\Box K_{1,r^{\prime}})\setminus X$ is a super dominating set of $K_{1,r}\square K_{1,r^{\prime}}$ with cardinality $(r+1)(r^{\prime}+1)-(r+r^{\prime})=rr^{\prime}+1$, and thus $\gamma_{\rm sp}(K_{1,r}\Box K_{1,r^{\prime}})\leq rr^{\prime}+1$. Therefore, $\gamma_{\rm sp}(K_{1,r}\Box K_{1,r^{\prime}})=rr^{\prime}+1$. ∎

Acknowledgements

This research was supported in part by the Spanish government under the grants MTM2016-78227-C2-1-P and PRX17/00102.