Affiliations: [a] Department of Mathematics, Faculty of Science, Selçuk University, Konya, Turkey
| [b] Department of Mathematics, Faculty of Science and Letters, Agri Ibrahim Cecen University, Ağri, Turkey
Correspondence:
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Corresponding author. Bünyamin Şahin, Department of Mathematics, Faculty of Science, Selçuk University, Konya, Turkey. E-mail: [email protected].
Abstract: In a graph G, a vertex v is dominated by an edge e, if e is incident with v or e is incident with a vertex which is a neighbor of v. An edge-vertex dominating set D is a subset of the edge set of G such that every vertex of G is edge-vertex dominated by an edge of D. The ev-domination number equals to the number of an edge-vertex dominating set of G which has minimum cardinality and it is denoted by γev (G). We here analyze double edge-vertex domination such that a double edge-vertex dominating set D is a subset of the edge set of G, provided that all vertices in G are ev-dominated by at least two edges of D. The double ev-domination number equals to the number of an double edge-vertex dominating set of G which has minimum cardinality and it is denoted by γdev (G). We demonstrate that the enumeration of the double ev-domination number of chordal graphs is NP-complete. Moreover several results about total domination number and double ev-domination number are obtained for trees.
Keywords: Trees, edge-vertex domination, double edge-vertex domination, total domination, domination
