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Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction., Comment: 25 pages, 11 figures

The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$ in terms of the number of vertices of degree $\Delta(G)$ (if $G$ is Class 2), and the numbers of vertices of degree $\Delta(G)$ and ${\Delta(G)-1}$ (if $G$ is Class 1). If $G$ is bipartite we give an exact expression for $\mathrm{es}_{\chi'}(G)$ involving the maximum size of a matching in the subgraph induced by vertices of degree $\Delta(G)$. Finally, we consider whether a minimum mitigating set, that is a set of size $\mathrm{es}_{\chi'}(G)$ whose removal reduces the chromatic index, has the property that every edge meets a vertex of degree at least $\Delta(G)-1$; we prove that this is true for some minimum mitigating set of $G$, but not necessarily for every minimum mitigating set of $G$., Comment: 10 pages, 4 figures

The chromatic vertex (resp.\ edge) stability number ${\rm vs}_{\chi}(G)$ (resp.\ ${\rm es}_{\chi}(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $\chi(H)=\chi(G)-1$. In the main result it is proved that if $G$ is a graph with $\chi(G) \in \{ \Delta(G), \Delta(G)+1 \}$, then ${\rm vs}_{\chi}(G) = {\rm ivs}_{\chi}(G)$, where ${\rm ivs}_{\chi}(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $\chi(G) \le \frac{\Delta(G)+1}{2}$. It is proved that if $\chi(G) > \frac{\Delta(G)}{2}+1$, then ${\rm vs}_{\chi}(G) = {\rm es}_{\chi}(G)$. A Nordhaus-Gaddum-type result on the chromatic vertex stability number is also given.

Let $G$ be a graph and $a(G)$, LIF$(G)$ denote the maximum orders of an induced forest and an induced linear forest of $G$, respectively. It is well-known that if $G$ is an $r$-regular graph of order $n$, then $a(G) \geq \frac{2}{r+1}n$. In this paper, we generalize this result by showing that LIF$(G) \geq \frac{2}{r+1}n$. It was proved that for every graph $G$, $a(G) \geq \sum_{i=1}^{n}\frac{2}{d_i+1}$, where $d_1, \ldots, d_n$ is the degree sequence of $G$. Here, we conjecture that for every graph $G$ with $\delta(G) \geq 2$, LIF$(G) \geq \sum_{i=1}^{n}\frac{2}{d_i+1}$., Comment: 9 pages, 1 algorithm, 2 tables

Let $G$ be a graph. The maximum nullity of $G$, denoted by $M(G)$, is defined to be the largest possible nullity over all real symmetric matrices $A$ whose $a_{ij}\neq 0$ for $i\neq j$, whenever two vertices $u_i$ and $u_j$ of $G$ are adjacent. In this paper, we characterize all cubic graphs with zero forcing number $3$. As a corollary, it is shown that if the zero forcing number is $3$, then $M(G)=3$. In addition, we introduce a family of cubic graphs containing graphs $G$ with $M(G)=Z(G)=4$. Also, we provide an algorithm which make a relation between maximum nullity of $G$ and the number of leaves in a spanning tree of $G$., Comment: 13 pages

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced vertex coloring $c : V (G) \rightarrow \mathbb{Z}_k$ defined by $c(v) = \sum_{e\in E_v} s(e) \mbox{ in } \mathbb{Z}_k,$ (where the indicated sum is computed in $\mathbb{Z}_k$ and $E_v$ denotes the set of all edges incident to $v$) results in a proper vertex coloring of $G$, then we refer to such a coloring as an improper twin $k$-edge coloring. The minimum $k$ for which $G$ has an improper twin $k$-edge coloring is called the improper twin chromatic index of $G$ and is denoted by $\chi'_{it}(G)$. In this paper, we show that if $G$ is a graph with vertex chromatic number $\chi(G)$, then $\chi'_{it}(G)=\chi(G)$, unless $\chi(G)=2 \pmod 4$ and in this case $\chi'_{it}(G)\in \{\chi(G), \chi(G)+1\}$. Moreover, we show that it is NP-hard to decide whether $\chi'_{it}(G)=\chi(G)$ or $\chi'_{it}(G)=\chi(G)+1$ and give some examples of perfect graph classes for which the problem is polynomial.

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if $G$ is a $2$-connected $k$-regular graph, $k \geq 3$, then every edge of $G$ is contained in an even cycle. We also prove that in a $2$-edge connected graph, if a vertex has odd degree, then there is an even cycle containing this vertex.

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and \gamma(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial., Comment: 13 pages