On vertex-edge and independent vertex-edge domination
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Given a graph G = (V,E), a vertex u ∈ V ve-dominates all edges incident to any vertex of N_G[u]. A set S ⊆ V is a ve-dominating set if for all edges e∈ E, there exists a vertex u ∈ 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.
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