Total $k$-rainbow domination subdivision number in graphs release_omfptjc6mnedfj3mprh5m3btca

by Rana Khoeilar, Mahla Kheibari, Zehui Shao, Seyed Mahmoud Sheikholeslami

Published in Computer Science Journal of Moldova by Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova.

2020   Volume 28, Issue 2(83), p152-169

Abstract

A total $k$-rainbow dominating function (T$k$RDF) of $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,\ldots,k\}$ such that (i) for any vertex $v\in V(G)$ with $f(v)=\emptyset$ the condition $\bigcup_{u \in N(v)}f(u)=\{1,\ldots,k\}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$, and (ii) the subgraph of $G$ induced by $\{v \in V(G) \mid f (v) \not =\emptyset\}$ has no isolated vertex. The total $k$-rainbow domination number, $\gamma_{trk}(G)$, is the minimum weight of a T$k$RDF on $G$. The total $k$-rainbow domination subdivision number ${\rm sd}_{\gamma_{trk}}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the total $k$-rainbow domination number. In this paper, we initiate the study of total $k$-rainbow domination subdivision number in graphs and we present sharp bounds for ${\rm sd}_{\gamma_{trk}}(G)$. In addition, we determine the total 2-rainbow domination subdivision number of complete bipartite graphs and show that the total 2-rainbow domination subdivision number can be arbitrary large.
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