Forbidden Subgraphs and Weak Locally Connected Graphs

A graph is called H-free if it has no induced subgraph isomorphic to H. A graph is called $$N^i$$Ni-locally connected if $$G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$$G[{x∈V(G):1≤dG(w,x)≤i}] is connected and $$N_2$$N2-locally connected if $$G[\{uv: \{uw, vw\}\subseteq E(G)\}]$$G[{uv:{uw,vw}⊆E(G)}] is connected for every vertex w of G, respectively. In this paper, we prove the following.Every 2-connected $$P_7$$P7-free graph of minimum degree at least three other than the Petersen graph has a spanning Eulerian subgraph. This implies that every H-free 3-connected graph (or connected $$N^4$$N4-locally connected graph of minimum degree at least three) other than the Petersen graph is supereulerian if and only if H is an induced subgraph of $$P_7$$P7, where $$P_i$$Pi is a path of i vertices.Every 2-edge-connected H-free graph other than $$\{K_{2, 2k+1}:k ~\text {is a positive integer}\}$${K2,2k+1:kis a positive integer} is supereulerian if and only if H is an induced subgraph of $$P_4$$P4.If every connected H-free $$N^3$$N3-locally connected graph other than the Petersen graph of minimum degree at least three is supereulerian, then H is an induced subgraph of $$P_7$$P7 or $$T_{2, 2, 3}$$T2,2,3, i.e., the graph obtained by identifying exactly one end vertex of $$P_3, P_3, P_4$$P3,P3,P4, respectively.If every 3-connected H-free $$N_2$$N2-locally connected graph is hamiltonian, then H is an induced subgraph of $$K_{1,4}$$K1,4. 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.