A dominating set $S$ of a graph $G(V,E)$ is called a \textit{secure dominating 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 \textit{secure domination number} $γ_s(G)$ of $G$ is the minimum cardinality of a secure dominating set of $G$. The \textit{Minimum Secure Domination problem} is to find a secure dominating set of a graph $G$ of cardinality $γ_s(G)$. 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 $(1-ε)~ln~|V|$ for any $ε> 0$, unless $NP \subseteq DTIME(|V|^{O(log~log~|V|)})$.

翻译：图$G(V,E)$的一个支配集$S$被称为\textit{安全支配集}，若对于每个顶点$u \in V(G) \setminus S$，存在一个相邻顶点$v \in S$，使得$(S \setminus \{v\}) \cup \{u\}$仍是$G$的一个支配集。$G$的\textit{安全支配数}$γ_s(G)$是$G$的最小安全支配集的基数。\textit{最小安全支配问题}旨在寻找图$G$中基数为$γ_s(G)$的安全支配集。本文研究了若干图类上安全支配问题的计算复杂性。安全支配问题的判定版本已被证明在星（梳）凸分裂图和Bisplit图上为NP完全。为此，我们进一步聚焦于Bisplit图在附加结构限制下的安全支配问题复杂性分析。特别地，通过引入弦性作为参数，我们分析了该参数对Bisplit图上问题计算状态的影响。我们建立了在Bisplit图环长限制下安全支配问题的P与NP-C二分状态。此外，我们证明了该问题在链图上可在多项式时间内求解。我们还证明，对于任意$ε> 0$，Bisplit图上的安全支配问题无法在$(1-ε)~ln~|V|$因子内近似，除非$NP \subseteq DTIME(|V|^{O(log~log~|V|)})$。