On Two Open Problems on Double Vertex-Edge Domination in Graphs

Fang Miao, Wenjie Fan, Mustapha Chellali, Rana Khoeilar, Seyed Mahmoud Sheikholeslami and Marzieh Soroudi
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Fang Miao: The Big Data Research Institute, Chengdu University, Chengdu 610106, China
Wenjie Fan: The Big Data Research Institute, Chengdu University, Chengdu 610106, China
Mustapha Chellali: LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270 Blida, Algeria
Rana Khoeilar: Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran
Seyed Mahmoud Sheikholeslami: Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran
Marzieh Soroudi: Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran

Mathematics, 2019, vol. 7, issue 11, 1-12

Abstract: A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v , as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S . The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G . A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V − S has at least two neighbors in S ). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G , and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

Keywords: vertex-edge dominating set; double vertex-edge dominating set; total dominating set; 2-dominating set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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