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

Given a graph G , we consider the Italian domination number γ I ( G ), the 2-rainbow domination number γ r 2 ( G ) and the Roman domination number γ R ( G ). It is known that γ I ( G ) ≤ γ r 2 ( G ) ≤ γ R ( G ) holds for any graph G . In this paper, we prove that γ I ( M ( G )) = γ r 2 ( M ( G )) = γ R ( 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.