Forbidden Subgraphs and Weak Locally Connected Graphs

Abstract

A graph is called H -free if it has no induced subgraph isomorphic to H. A graph is called $$N^i$$ -locally connected if $$G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$$ is connected and $$N_2$$ -locally connected if $$G[\{uv: \{uw, vw\}\subseteq E(G)\}]$$ is connected for every vertex w of G, respectively. In this paper, we prove the following. We present an algorithm to find a collapsible subgraph of a graph with girth 4 whose idea is used to prove our first conclusion above. Finally, we propose that the reverse of the last two items would be true.