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- W2608820967 abstract "Let $G=(V, E)$ be a graph. A set $Ssubseteq V(G)$ is a {it dominating set} of $G$ if every vertex in $Vsetminus S$ is adjacent to a vertex of $S$. The {it domination number} of $G$, denoted by $gamma(G)$, is the cardinality of a minimum dominating set of $G$. Furthermore, a dominating set $S$ is an {it independent transversal dominating set} of $G$ if it intersects every maximum independent set of $G$. The {it independent transversal domination number} of $G$, denoted by $gamma_{it}(G)$, is the cardinality of a minimum independent transversal dominating set of $G$. In 2012, Hamid initiated the study of the independent transversal domination of graphs, and posed the following two conjectures: Conjecture 1. If $G$ is a non-complete connected graph on $n$ vertices, then $gamma_{it}(G)leqlceilfrac{n}{2}rceil$. Conjecture 2. If G is a connected bipartite graph, then $gamma_{it}(G)$ is either $gamma(G)$ or $gamma(G)+1$. We show that Conjecture 1 is not true in general. Very recently, Conjecture 2 is partially verified to be true by Ahangar, Samodivkin, Yero. Here, we prove the full statement of Conjecture 2. In addition, we give a correct version of a theorem of Hamid. Finally, we answer a problem posed by Martinez, Almira, and Yero on the independent transversal total domination of a graph." @default.
- W2608820967 created "2017-05-05" @default.
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- W2608820967 date "2017-04-20" @default.
- W2608820967 modified "2023-09-27" @default.
- W2608820967 title "Independent transversal domination number of a graph" @default.
- W2608820967 hasPublicationYear "2017" @default.
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