Graphs with unique minimum edge-vertex dominating sets

Document Type : Original paper

Authors

1 Department of Mathematics, SASTRA Deemed to be University Thanjavur, Tamil Nadu, India

2 LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

Abstract

An edge $e$ of a simple graph $G=(V_{G},E_{G})$ is said to ev-dominate a vertex $v\in V_{G}$ if $e$ is incident with $v$ or $e$ is incident with a vertex adjacent to $v$. A subset $D\subseteq E_{G}$ is an edge-vertex dominating set (or an evd-set for short) of $G$ if every vertex of $G$ is ev-dominated by an edge of $D$. The edge-vertex domination number of $G$ is the minimum cardinality of an evd-set of $G$. In this paper, we initiate the study of the graphs with unique minimum evd-sets that we will call UEVD-graphs. We first present some basic properties of UEVD-graphs, and then we characterize UEVD-trees by equivalent conditions as well as by a constructive method.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 11 September 2023

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