Secure domination number of kk-subdivision of graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called a secure dominating set of $G$, if for every $u\in V-D$, there exists a vertex $v\in D$ such that $uv \in E$ and $D-\{v\}\cup\{u\}$ is a dominating set of $G$. The cardinality of a smallest secure dominating set of $G$, denoted by $\gamma_s(G)$, is the secure domination number of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the secure domination number of $k$-subdivision of $G$.Comment: 10 Pages, 8 Figure

On the Complexity of Co-secure Dominating Set Problem

A set $D \subseteq V$ of a graph $G=(V, E)$ is a dominating set of $G$ if every vertex $v\in V\setminus D$ is adjacent to at least one vertex in $D.$ A set $S \subseteq V$ is a co-secure dominating set (CSDS) of a graph $G$ if $S$ is a dominating set of $G$ and for each vertex $u \in S$ there exists a vertex $v \in V\setminus S$ such that $uv \in E$ and $(S\setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. The minimum cardinality of a co-secure dominating set of $G$ is the co-secure domination number and it is denoted by $\gamma_{cs}(G)$. Given a graph $G=(V, E)$, the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of $(1- \epsilon)\ln |V|$ for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of $O(\ln |V|)$ for any graph $G$ with $\delta(G)\geq 2$. For $3$-regular and $4$-regular graphs, we show that Min Co-secure Dom is approximable within a factor of $\dfrac{8}{3}$ and $\dfrac{10}{3}$, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for $3$-regular graphs.Comment: 12 pages, 2 figure