On Vertex-Edge and Independent Vertex-Edge Domination

Abstract

Given a graph G = (V, E), a vertex u ∈ V ve-dominates all edges incident to any vertex of NG [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 vedominates 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.

Edge-Vertex Dominating Set in Unit Disk Graphs

“…Finding even in cubic planar graphs is NP-hard [6]. The vertex-edge domination problem can be solved in linear time on block graphs [7]. In the same paper it is also shown that finding in undirected path graphs is NP-complete.…”

Vertex-edge domination in interval and bipartite permutation graphs

“…Approximation algorithm and approximation hardness results are proved in [14]. In [16], the authors designed a linear-time algorithm for the ve-domination problem in block graphs and showed that this problem remains NP-complete for undirected path graphs. Recently, Jena and Das [9] have studied the ve-domination problem in unit disk graphs.…”