Inflated graphs with equal independence number and upper irredundance number Original Research Article

The inflation GI of a graph G is obtained from G by replacing each vertex x of degree d(x) by a clique X≃Kd(x) and each edge xy by an edge between two vertices of the corresponding cliques X and Y of GI in such a way that the edges of GI which come from the edges of G form a matching of GI. Some properties related to the parameters of independence, domination and irredundance of an inflation GI have already been studied in Dunbar and Haynes (Congr. Numer. 118 (1996) 143), Favaron (J. Graph Theory 28 (2) (1998) 97) and Puech (J. Combin. Math. Combin. Comput. 33 (2000) 117–127). We prove here that if we denote by β and IR the independence number and the upper irredundance number of a graph, the 2-connected graphs G satisfying β(GI)=IR(GI) are those ones for which maxcut(G) is at most the order of G (these graphs have been determined in Delorme and Favaron (Utilitas Math. 56 (1999) 153)).