Paper
Vertex-edge domination on subclasses of bipartite graphs
Authors
Arti Pandey, Kaustav Paul, Kamal Santra
Abstract
Given a simple undirected graph , the open neighbourhood of a vertex is defined as , and the closed neighbourhood as . A subset is called a vertex-edge dominating set if, for every edge , at least one vertex from appears in ; that is, . Intuitively, a vertex-edge dominating set ensures that every edge, as well as all edges incident to either of its endpoints, is dominated by at least one vertex from the set. The \textsc{Min-VEDS} problem asks for a vertex-edge dominating set of minimum size in a given graph. This problem is known to be NP-complete even for bipartite graphs. In this paper, we strengthen this hardness result by proving that the problem remains NP-complete for two specific subclasses of bipartite graphs: star-convex and comb-convex bipartite graphs. For a graph on vertices, it is known that the \textsc{Min-VEDS} problem cannot be approximated within a factor of for any , unless . We also prove that this inapproximability result holds even for star-convex and comb-convex bipartite graphs. On the positive side, we present a polynomial-time algorithm for computing a minimum vertex-edge dominating set in convex bipartite graphs. A polynomial-time algorithm for this graph class was also proposed by B{ü}y{ü}k{ç}olak et al.~\cite{buyukccolak2025linear}, but we show that their algorithm has certain flaws by providing instances where it fails to produce an optimal solution. We address this issue by presenting a modified algorithm that correctly computes an optimal solution.