Vertex-edge domination on subclasses of bipartite graphs

2025-12-13

AI Summary

This paper investigates the Minimum Vertex-Edge Dominating Set (Min-VEDS) problem on subclasses of convex bipartite graphs. First, it establishes that Min-VEDS remains NP-complete on star-convex and comb-convex bipartite graphs—its first such hardness proof—and proves strong inapproximability: no polynomial-time constant-factor approximation exists unless P = NP. Second, it identifies counterexamples and correctness flaws in prior algorithms for convex bipartite graphs, and proposes a rigorously correct linear-time exact algorithm with O(n + m) time complexity. The main contributions are: (i) settling the computational complexity boundary of Min-VEDS on two fundamental convex graph classes; and (ii) providing the first efficient, verifiably optimal algorithm for Min-VEDS on convex bipartite graphs, thereby resolving a longstanding algorithmic gap for this problem under convex structural constraints.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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, $vert (N_G[u] cup N_G[v]) cap Dvert geq 1$. 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 extsc{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 extsc{Min-VEDS} problem cannot be approximated within a factor of $(1 - ε)ln |V|$ for any $ε> 0$, unless $ ext{NP} subseteq ext{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{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.

Problem

• Proves NP-completeness of Min-VEDS on star-convex and comb-convex bipartite graphs
• Establishes inapproximability results for these bipartite graph subclasses
• Provides corrected polynomial algorithm for convex bipartite graphs

Innovation

• Proved NP-completeness for star-convex bipartite graphs
• Proved NP-completeness for comb-convex bipartite graphs
• Designed polynomial-time algorithm for convex bipartite graphs