Total vertex-edge domination

ABSTRACT

A vertex v of a graph $G=(V,E)$ is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is a vertex-edge dominating set (or simply, a ve-dominating set) if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a ve-dominating set of G is the vertex-edge domination number ${\gamma }_{ve}\left(G\right).$ A ve-dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total ve-dominating set of G is the total vertex-edge domination number ${\gamma }_{ve}^t\left(G\right).$ In this paper we initiate the study of total vertex-edge domination. We show that determining the number ${\gamma }_{ve}^t\left(G\right)$ for bipartite graphs is NP-complete. Then we show that if T is a tree different from a star with order n, ℓ leaves and s support vertices, then ${\gamma }_{ve}^t\left(T\right)\le (n-\ell +s)/2.$ Moreover, we characterize the trees attaining this upper bound. Finally, we establish a necessary condition for graphs G such that ${\gamma }_{ve}^t\left(G\right)=2{\gamma }_{ve}\left(G\right)$ and we provide a characterization of all trees T with ${\gamma }_{ve}^t\left(T\right)=2{\gamma }_{ve}\left(T\right)$.

KEYWORDS:

• Total vertex-edge domination
• vertex-edge domination
• total domination
• domination
• tree

2000 MSC SUBJECT CLASSIFICATION:

• 05C69

Disclosure statement

No potential conflict of interest was reported by the authors.