LAUSR

Let $$\mathcal {R}(G,H)$$R(G,H) denote the set of all graphs F satisfying $$F \rightarrow (G,H)$$F→(G,H) and for every $$e \in E(F),$$e∈E(F),$$(F-e) \nrightarrow (G,H).$$(F-e)↛(G,H). In this paper, we derive the necessary and sufficient conditions for graphs belonging to $$\mathcal {R}(mK_2,H)$$R(mK2,H) for any graph H and each positive integer m. We give all disconnected graphs in $$\mathcal {R}(mK_2,H),$$R(mK2,H), for any connected graph H. Furthermore, we prove that if $$F \in \mathcal {R}(mK_2,P_3),$$F∈R(mK2,P3), then any graph obtained by subdividing one non-pendant edge in F will be in $$\mathcal {R}((m+1)K_2,P_3)$$R((m+1)K2,P3).