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Results on vertex-edge and independent vertex-edge domination
Journal of Combinatorial Optimization ( IF 0.9 ) Pub Date : 2021-12-01 , DOI: 10.1007/s10878-021-00832-z
Subhabrata Paul 1 , Keshav Ranjan 2
Affiliation  

Given a graph \(G = (V,E)\), a vertex \(u \in V\) ve-dominates all edges incident to any vertex of \(N_G[u]\). A set \(S \subseteq V\) is a ve-dominating set if for all edges \(e\in E\), there exists a vertex \(u \in S\) such that u ve-dominates e. Lewis (Vertex-edge and edge-vertex parameters in graphs. Ph.D. thesis, Clemson, SC, USA, 2007) proposed a linear time algorithm for ve-domination problem for trees. In this paper, we have constructed an example where the algorithm proposed by Lewis, fails. We have proposed linear time algorithms for ve-domination and independent ve-domination problem in block graphs, which is a superclass of trees. We have also proposed a linear time algorithm for weighted ve-domination problem in trees. We have also proved that finding minimum ve-dominating set is NP-complete for undirected path graphs. Finally, we have characterized the trees with equal ve-domination and independent ve-domination number.



中文翻译:

顶点边缘和独立顶点边缘支配的结果

给定一个图\(G = (V,E)\),顶点\(u \in V\) ve 支配\(N_G[u]\) 的任何顶点相关的所有边。集合\(S \subseteq V\)是一个ve 支配集,如果对于所有边\(e\in E\),都存在一个顶点\(u \in S\)使得u ve-dominates e. Lewis(图中的顶点-边和边-顶点参数。博士论文,克莱姆森,南卡罗来纳州,美国,2007 年)提出了一种用于树的 ve-domination 问题的线性时间算法。在本文中,我们构建了一个示例,其中 Lewis 提出的算法失败。我们已经提出了线性时间算法来解决块图中的 ve-domination 和独立 ve-domination 问题,这是树的超类。我们还为树中的加权 ve-domination 问题提出了一种线性时间算法。我们还证明,对于无向路径图,找到最小 ve 支配集是 NP 完全的。最后,我们对具有相等 ve-domination 和独立 ve-domination 数的树进行了表征。

更新日期:2021-12-03
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