Outer-independent total 2-rainbow dominating functions in graphs

Communications in Combinatorics and Optimization

article

Akram Mahmoodi¹  Lutz Volkmann² 

[1] Department of Mathematics Payame Noor University I.R. Iran;RWTH Aachen University

关键词: Domination number;    $2$-rainbow domination number;    total $2$-rainbow domination number;    outer-independent total $2$-rainbow domination number;   

DOI  :  10.22049/cco.2022.27753.1344

学科分类：社会科学、人文和艺术（综合）

来源: Azarbaijan Shahide Madani Universit

【 摘 要 】

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. An {outer-independent total $2$-rainbow dominating function of a graph $G$ is a function $f$ from $V(G)$ to the set of all subsets of $\{1,2\}$ such that the following conditions hold: (i) for any vertex $v$ with $f(v)=\emptyset$ we have $\bigcup_{u\in N_G(v)} f(u)=\{1,2\}$, (ii) the set of all vertices $v\in V(G)$ with $f(v)=\emptyset$ is independent and (iii) $\{v\mid f(v)\neq\emptyset\}$ has no isolated vertex. The outer-independent total $2$-rainbow domination number of $G$, denoted by ${\gamma}_{oitr2}(G)$, is the minimum value of $\omega(f)=\sum_{v\in V(G)} |f(v)|$ over all such functions $f$. In this paper, we study the outer-independent total $2$-rainbow domination number of $G$ and classify all graphs with outer-independent total $2$-ainbow domination number belonging to the set $\{2,3,n\}$. Among other results, we present some sharp bounds concerning the invariant.

【 授权许可】

CC BY-SA