Characterizing the Difference Between Graph Classes Defined by Forbidden Pairs Including the Claw

For two graphs A and B , a graph G is called {A,B} -free if G contains neither A nor B as an induced subgraph. Let P_n denote the path of order n . For nonnegative integers k , ℓ and m , let N_k,ℓ ,m be the graph obtained from K_3 and three vertex-disjoint paths P_k+1 , P_ℓ +1 , P_m+1 by identifying each of the vertices of K_3 with one endvertex of one of the paths. Let Z_k=N_k,0,0 and B_k,ℓ=N_k,ℓ ,0 . Bedrossian characterized all pairs {A,B} of connected graphs such that every 2-connected {A,B} -free graph is Hamiltonian. All pairs appearing in the characterization involve the claw ( K_1,3 ) and one of N_1,1,1 , P_6 and B_1,2 . In this paper, we characterize connected graphs that are (i) {K_1,3,Z_2} -free but not B_1,1 -free, (ii) {K_1,3,B_1,1} -free but not P_5 -free, or (iii) {K_1,3,B_1,2} -free but not P_6 -free. The third result is closely related to Bedrossian’s characterization. Furthermore, we apply our characterizations to some forbidden pair problems.