Hamiltonicity of complements of middle graphs

For a graph G  , the middle graph M(G)M(G) of G   is the graph with vertex set V(G)∪E(G)V(G)∪E(G) in which the vertices x and y   are joined by an edge if {x,y}∩E(G)≠∅{x,y}∩E(G)≠∅, and x and y are adjacent or incident in G  . In this note, we show that the complement of middle graph M(G)M(G) of a graph G is hamiltonian if and only if G   is not a star and is not isomorphic to any graph in {K1,2K1,K2,K2∪K1,K3,K3∪K1}{K1,2K1,K2,K2∪K1,K3,K3∪K1}.