Difference between [formula omitted]-rainbow domination and Roman domination in graphs

Abstract

A 2-rainbow dominating functionf of a graph G is a function f:V(G)→2{1,2} such that, for each vertex v∈V(G) with f(v)=0̸, ⋃u∈NG(v)f(u)={1,2}. The minimum of ∑v∈V(G)|f(v)| over all such functions is called the 2-rainbow domination numberγr2(G). A Roman dominating functiong of a graph G, is a function g:V(G)→{0,1,2} such that, for each vertex v∈V(G) with g(v)=0, v is adjacent to a vertex u with g(u)=2. The minimum of ∑v∈V(G)g(v) over all such functions is called the Roman domination numberγR(G).Regarding 0̸ as 0, these two dominating functions have a common property that the same three integers are used and a vertex having 0 must be adjacent to a vertex having 2. Motivated by this similarity, we study the relationship between γR(G) and γr2(G). We also give some sharp upper bounds on these dominating functions. Moreover, one of our results tells us the following general property in connected graphs: any connected graph G of order n≥3 contains a bipartite subgraph H=(A,B) such that δ(H)≥1 and |A|−|B|≥n/5. The bound on |A|−|B| is best possible.