On vertex-edge and independent vertex-edge domination

TL;DR

This paper investigates ve-domination in graphs, identifies flaws in existing algorithms for trees, proposes a new linear-time algorithm for block graphs, proves NP-completeness for path graphs, and characterizes trees with equal ve-domination and independent ve-domination numbers.

Contribution

It corrects previous algorithmic assumptions, introduces a new linear-time algorithm for block graphs, and characterizes specific tree properties related to ve-domination.

Findings

Existing algorithm fails on certain trees.

New linear-time algorithm for ve-domination in block graphs.

NP-completeness of minimum ve-dominating set in path graphs.

Abstract

Given a graph $G = (V,E)$, a vertex $u \in V$ ve-dominates all edges incident to any vertex of $N_G[u]$. A set $S \subseteq V$ is a ve-dominating set if for all edges $e\in E$, there exists a vertex $u \in S$ such that $u$ ve-dominates $e$. Lewis [Ph.D. thesis, 2007] proposed a linear time algorithm for ve-domination problem for trees. In this paper, first we have constructed an example where the proposed algorithm fails. Then we have proposed a linear time algorithm for ve-domination problem in block graphs, which is a superclass of trees. We have also proved that finding minimum ve-dominating set is NP-complete for undirected path graphs. Finally, we have characterized the trees with equal ve-domination and independent ve-domination number.