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}，当且仅当对于每个顶点$u \in V(G) \setminus S$，均存在相邻顶点$v \in S$使得$(S \setminus \{v\}) \cup \{u\}$构成$G$的支配集。图$G$的\textit{安全支配数}$γ_s(G)$是其安全支配集的最小基数。\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。本文研究了安全支配问题在若干图类上的计算复杂性。该问题的判定版本已被证明在星型（梳状）凸分裂图与双分裂图上具有NP完全性。因此，我们进一步聚焦于在双分裂图上附加结构约束后的安全支配问题复杂性分析。特别地，通过引入弦图性作为参数，我们分析了该参数对双分裂图上问题计算状态的影响。我们在双分裂图的环长约束下建立了安全支配问题的P类与NP完全类的二分判定状态。此外，我们证明了该问题在链图中具有多项式时间可解性。同时，我们证明了对于任意$ε> 0$，除非$NP \subseteq DTIME(|V|^{O(log~log~|V|)})$成立，否则双分裂图上的安全支配问题不存在近似比为$(1-ε)~ln~|V|$的近似算法。