Italian, 2-rainbow and Roman domination numbers in middle graphs

Abstract

Given a graph G, we consider the Italian domination number γI(G), the 2-rainbow domination number γr2(G) and the Roman domination number γR(G). It is known that γI(G) ≤ γr2(G) ≤ γR(G) holds for any graph G. In this paper, we prove that γI(M(G)) = γr2(M(G)) = γR(M(G)) = n for the middle graph M(G) of a graph G of order n, which gives an answer for an open problem posed by Chellali et al. [Discrete Appl. Math. 204 (2016) 22–28]. Moreover, we give a complete characterization of Roman domination stable middle graphs, 2-rainbow domination stable middle graphs and Italian domination stable middle graphs.