Linear time algorithm for the vertex-edge domination problem in convex bipartite graphs

Given a graph $G=(V,E)$, a vertex $u\in V$ ve-dominates all edges incident to any vertex in the closed neighborhood $N\left[u\right]$. A subset $D\subseteq V$ is a vertex-edge dominating set if, for each edge $e\in E$, there exists a vertex $u\in D$ such that $u$ ve-dominates $e$. The objective of the ve-domination problem is to find a minimum cardinality ve-dominating set in $G$. In this paper, we present a linear time algorithm to find a minimum cardinality ve-dominating set for convex bipartite graphs, which is a superclass of bipartite permutation graphs and a subclass of bipartite graphs, where the ve-domination problem is solvable in linear time and NP-complete, respectively. We also establish the relationship ${\gamma }_{ve}=i_{ve}$ for convex bipartite graphs. Our approach leverages a chain decomposition of convex bipartite graphs, allowing for efficient identification of minimum ve-dominating sets and extending algorithmic insights into ve-domination for specific structured graph classes.