Algorithm for finding vertex-edge domination number on graphs with bounded treewidth and related problems on planar graphs

TL;DR

This paper introduces a polynomial-time algorithm for computing the vertex-edge domination number on graphs with bounded treewidth and explores related problems on planar graphs, improving understanding of these parameters.

Contribution

It presents a new polynomial-time algorithm for ve-domination on bounded treewidth graphs and establishes a bound on treewidth relative to ve-domination number in planar graphs.

Findings

Polynomial-time algorithm for ve-domination on graphs with bounded treewidth.

Treewidth of planar graphs is O(√ve-domination number).

An O(c^{√k}|V(G)|) algorithm for k-ve-domination on planar graphs.

Abstract

Given a graph $G=(V,E)$, a vertex $u \in V$ {\em ve-dominates} all edges incident to any vertex of $N_G[u]$. A set $S \subseteq V$ is a {\em ve-dominating set} if for all edges $e\in E$, there exists a vertex $u\in S$ such that $u$ ve-dominates $e$. The minimum cardinality among all ve-dominating sets is known as the \textit{vertex-edge domination number} (or simply ve-domination number) and denoted by $\gamma_{ve}(G)$. Finding a minimum ve-dominating set was proved to be NP-complete. Restricted to trees, the problem admits a linear-time algorithm. Treewidth is a commonly used parameter for solving NP-hard problems. In this paper, we present a polynomial-time algorithm for finding a minimum ve-dominating set on graphs with bounded treewidth. Moreover, we show that the treewidth of a planar graph $G$ with ve-domination number $\gamma_{ve}(G)$ is $O(\sqrt{\gamma_{ve}(G)})$ and present an $O(c^{\sqrt{k}}|V(G)|)$-time algorithm for the $k$-ve-domination problem on planar graphs.