# Dominating Vertex Covers: The Vertex-Edge Domination Problem

## Document Type

Article

## Publication Date

2-1-2020

## Abstract

The vertex-edge domination number of a graph, γve(G), is defined to be the cardinality of a smallest set D such that there exists a vertex cover C of G such that each vertex in C is dominated by a vertex in D. This is motivated by the problem of determining how many guards are needed in a graph so that a searchlight can be shone down each edge by a guard either incident to that edge or at most distance one from a vertex incident to the edge. Our main result is that for any cubic graph G with n vertices, γve(G) ≤ 9n/26. We also show that it is NP-hard to decide if γve(G) = γ(G) for bipartite graph G.

## Publication Title

Discussiones Mathematicae - Graph Theory

## Volume

41

## Issue

1

## First Page

123

## Last Page

132

## Digital Object Identifier (DOI)

10.7151/dmgt.2175

## ISSN

12343099

## E-ISSN

20835892

## Citation Information

Klostermeyer, W.F., Messinger, M.E., Yeo, A. (2020) Dominating Vertex Covers: The Vertex-Edge Domination Problem. Discussiones Mathematicae - Graph Theory, 41(1), 123-132.