Secure Domination in Bisplit graphs -- A Structural and algorithmic study

A dominating set $S$ of a graph $G(V,E)$ is called a $\mathit{\text{secure}}\mathit{\text{dominating}}\mathit{\text{set}}$ if each vertex $u\in V(G)\setminus S$ is adjacent to a vertex $v\in S$ such that $(S\setminus \{v\})\cup \{u\}$ is a dominating set of $G$. The $\mathit{\text{secure}}\mathit{\text{domination}}\mathit{\text{number}}$ ${\gamma }_s(G)$ of $G$ is the minimum cardinality of a secure dominating set of $G$. The $\mathit{\text{Minimum}}\mathit{\text{Secure}}\mathit{\text{Domination}}\mathit{\text{problem}}$ is to find a secure dominating set of a graph $G$ of cardinality ${\gamma }_s(G)$. In this paper, the computational complexity of the secure domination problem on several graph classes is investigated. The decision version of secure domination problem was shown to be NP-complete on star(comb) convex split graphs and bisplit graphs. So we further focus on complexity analysis of secure domination problem under additional structural restrictions on bisplit graphs. In particular, by imposing chordality as a parameter, we analyse its impact on the computational status of the problem on bisplit graphs. We establish the P versus NP-C dichotomy status of secure domination problem under restrictions on cycle length within bisplit graphs. In addition, we establish that the problem is polynomial-time solvable in chain graphs. We also prove that the secure domination problem cannot be approximated for a bisplit graph within a factor of $(1-\epsilon )ln\mathrm{\mid }V\mathrm{\mid }$ for any $\epsilon >0$, unless $NP\subseteq DTIME(\mathrm{\mid }V{\mathrm{\mid }}^{O(loglog\mathrm{\mid }V\mathrm{\mid })})$.