论文标题
图形中某些统治变体的算法方面
Algorithmic Aspects of Some Variants of Domination in Graphs
论文作者
论文摘要
一个$ s \ subseteq v $是G中的一个主体集，如果对于v \ s中的每个u \，则在s $中存在$ v \，因此$（u，v）\ in E $，即$ n [s] = v $。如果诱导子图$ g [s] $至少具有一个孤立的顶点，则主导集合$ s $是一个孤立统治集}（ids）。众所周知，分离株统治决策问题（IDOM）对于二分图是NP的组件。在本文中，我们通过证明IDOM是分裂图和完美的消除两分图的NP完整的，这是两部分图的子类。如果G [S]没有边缘，则设置$ S \ subseteq V $是独立的集合。如果对于v \ setminus s $中的每个顶点$ u \，则S s s \ subseteq V是$ g $的安全主导集，在s $中存在一个顶点$ v \，以至于$（u，v）\ in E $和$（s \ \ \ \ \ \ \ {v \}）此外，我们启动了一种称为独立安全统治的新的支配参数的研究。如果$ s $是独立的集合，并且是$ g $的安全主导集，则设置$ s \ subseteq v $是独立的安全主导集（INSDS）。 $ g $的insds的最小尺寸称为$ g $的独立安全统治数，并用$γ_{is}（g）$表示。给定图形$ g $和一个正整数$ k，$ insdm问题是检查$ g $最多具有$k。$k。$ $k。$ g。我们证明，对于两极分子的图形和可求解的界限时间和线性时间，用于有限的树宽度图和threshold图形图形图和threshold图，是一个分裂图的子类别的线性时间。 MINSDS问题是在输入图中找到一个最小尺寸的独立安全统治集。最后，我们证明了Minsds问题是最高度量的图表的APX-HARD。
A set $S \subseteq V$ is a dominating set in G if for every u \in V \ S, there exists $v \in S$ such that $(u,v) \in E$, i.e., $N[S] = V$. A dominating set $S$ is an Isolate Dominating Set} (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set $S \subseteq V$ is an independent set if G[S] has no edge. A set S \subseteq V is a secure dominating set of $G$ if, for each vertex $u \in V \setminus S$, there exists a vertex $v \in S$ such that $ (u,v) \in E $ and $(S \ \{v\}) \cup \{u\}$ is a dominating set of $G$. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set $S\subseteq V$ is an Independent Secure Dominating Set (InSDS) if $S$ is an independent set and a secure dominating set of $G$. The minimum size of an InSDS in $G$ is called the independent secure domination number of $G$ and is denoted by $γ_{is}(G)$. Given a graph $ G$ and a positive integer $ k,$ the InSDM problem is to check whether $ G $ has an independent secure dominating set of size at most $ k.$ We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we prove that the MInSDS problem is APX-hard for graphs with maximum degree $5.$
