The rainbow domination subdivision numbers of graph

Abstract

{$2$-rainbow dominating function} (2RDF) of a graph $G$ isa function $f$ from the vertex set $V(G)$ to the set of allsubsets of the set $\{1,2\}$ such that for any vertex $vınV(G)$with $f(v)=\mathrm{\varnothing }$ the condition ${\cup }_{uınN(v)}f(u)=\{1,2\}$ is fulfilled. The {weight} of a 2RDF $f$ isthe value $\omega (f)={\mathrm{\Sigma }}_{vınV}|f(v)|$. The {$2$-rainbowdomination number} of a graph $G$, denoted by ${\gamma }_{r2}(G)$, isthe minimum weight of a 2RDF of G. The {$2$-rainbow dominationsubdivision number} $\text{tx}{\mathrm{s}\mathrm{d}}_{{\gamma }_{r2}}(G)$ is the minimumnumber of edges that must be subdivided (each edge in $G$ can besubdivided at most once) in order to increase the $2$-rainbowdomination number. In this paper, we initiate the study of$2$-rainbow domination subdivision number in graphs.