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

by Fang Miao 1,2, Wenjie Fan 1,2, Mustapha Chellali 3, Rana Khoeilar 4, Seyed Mahmoud Sheikholeslami 4,* and Marzieh Soroudi 4
1
The Big Data Research Institute, Chengdu University, Chengdu 610106, China
2
Key Laboratory of Pattern Recognition and Intelligent Information Processing, Chengdu 610106, China
3
LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270 Blida, Algeria
4
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 51368, Iran
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1010; https://doi.org/10.3390/math7111010
Received: 13 September 2019 / Revised: 18 October 2019 / Accepted: 21 October 2019 / Published: 24 October 2019
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 $γ d v e ( T ) = γ 2 ( T )$ . View Full-Text
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Miao, F.; Fan, W.; Chellali, M.; Khoeilar, R.; Sheikholeslami, S.M.; Soroudi, M. On Two Open Problems on Double Vertex-Edge Domination in Graphs. Mathematics 2019, 7, 1010.