The Local Structure of Claw-Free Graphs Without Induced Generalized Bulls

Abstract

In this paper, we show the following: Let G be a connected claw-free graph such that G has a connected induced subgraph H that has a pair of vertices $$\{v_{1}, v_{2}\}$$ of degree one in H whose distance is $$d + 2$$ in H. Then H has an induced subgraph F, which is isomorphic to $$B_{i,j}$$ , with $$\{v_{1}, v_{2}\} \subseteq V(F)$$ and $$i+j=d+1$$ , with a well-defined exception. Here $$B_{i, j}$$ denotes the graph obtained by attaching two vertex-disjoint paths of lengths $$i, j \ge 1$$ to a triangle. We also use the result above to strengthen the results in Xiong et al. (Discrete Math 313:784–795, 2013) in two cases, when $$i + j \le 9$$ , and when the graph is $$\Gamma _{0}$$ -free. Here $$\Gamma _{0}$$ is the simple graph with degree sequence 4, 2, 2, 2, 2. Let $$i, j > 0$$ be integers such that $$i + j \le 9$$ . Then The two results above are all sharp in the sense that the condition “ $$i+j\le 9$$ ” couldn’t be replaced by $$``i+j\le 10$$ ”.