Vertex-edge domination on subclasses of bipartite graphs

Abstract

Given a simple undirected graph $G = (V, E)$, the open neighbourhood of a vertex $v \in V$ is defined as $N_G(v) = \{u \in V \mid uv \in E\}$, and the closed neighbourhood as $N_G[v] = N_G(v) \cup \{v\}$. A subset $D \subseteq V$ is called a vertex-edge dominating set if, for every edge $uv \in E$, at least one vertex from $D$ appears in $N_G[u] \cup N_G[v]$; that is, $\vert (N_G[u] \cup N_G[v]) \cap D\vert \geq 1$. 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 $G$ on $n$ vertices, it is known that the \textsc{Min-VEDS} problem cannot be approximated within a factor of $(1 - \epsilon)\ln |V|$ for any $\epsilon > 0$, unless $\text{NP} \subseteq \text{DTIME}(|V|^{O(\log \log |V|)})$. 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{\"u}y{\"u}k{\c{c}}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.