All $$M_{2}$$ M 2 -Equicoverable Graphs

Abstract

A graph G is called H-equicoverable if every minimal H-covering in G is also a minimum H-covering in G. All $$P_{3}$$P3-equicoverable graphs were characterized in Zhang (Discret Appl Math 156:647---661, 2008). In 2011, connected $$M_2$$M2-equicoverable graphs that contains cycles were characterized [see Zhang and Jiang (J Tianjin Univ:44:466---470, 2011) and Zhang and Zhang (Ars Comb 101:45---63, 2011)]. In this paper, we give the characterization of all disconnected $$M_{2}$$M2-equicoverable graphs and all $$M_{2}$$M2-equicoverable trees. So all $$M_2$$M2-equicoverable graphs are characterized now.