Well ve-covered graphs

Document Type : Original paper


1 Faculty of Economic Sciences and Management. University of Boumerdes, Algeria

2 Department of Mathematics University of Blida B.P. 270, Blida, Algeria


A vertex $u$ of a graph $G=(V,E)$ ve-dominates every edge incident to $u$ as well as every edge adjacent to these incident edges. A set $S\subseteq V$ is a vertex-edge dominating set (or a ved-set for short) if every edge of $E$ is ve-dominated by at least one vertex in $S$. A ved-set is independent if its vertices are pairwise non-adjacent. The independent ve-domination number $i_{ve}(G)$ is the minimum cardinality of an independent ved-set and the upper independent ve-domination number $\beta_{ve}(G)$ is the maximum cardinality of a minimal independent ved-set of $G$. In this paper, we are interesting in graphs $G$ such that $i_{ve}(G)=\beta_{ve}(G)$, which we call well ve-covered graphs. We show that recognizing well ve-covered graphs is co-NP-complete, and we present a constructive characterization of well ve-covered trees.


Main Subjects

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