TY - JOUR
AU - Amjadi, Jafar
AU - Khoeilar, Rana
AU - Dehgardi, N.
AU - Volkmann, Lutz
AU - Sheikholeslami, S.M.
PY - 2018/06/30
Y2 - 2022/05/24
TI - The restrained rainbow bondage number of a graph
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 49
IS - 2
SE -
DO - 10.5556/j.tkjm.49.2018.2365
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/2365
SP - 115-127
AB - A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the restrained $k$-rainbow domination number of $G$, denoted by $\gamma_{rrk}(G)$. The restrained $k$-rainbow bondage number $b_{rrk}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $F \subseteq E(G)$ for which $\gamma_{rrk}(G-F) > \gamma_{rrk}(G)$. In this paper, we initiate the study of the restrained $k$-rainbow bondage number in graphs and we present some sharp bounds for $b_{rr2}(G)$. In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.
ER -