Abstract
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 [Ph.D. thesis, 2007] proposed a linear time algorithm for ve-domination problem for trees. In this paper, first we have constructed an example where the proposed algorithm fails. Then we have proposed a linear time algorithm for ve-domination problem in block graphs, which is a superclass of 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.
Keywords
K. Ranjan—This work was done when the second author was persuing his M.Tech. at IIT Patna.
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Paul, S., Ranjan, K. (2019). On Vertex-Edge and Independent Vertex-Edge Domination. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_35
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