On the structure of pentagraphs

•Let G∈G,u∈V(G), and B be a component of G[Ll(u)]. Then B is a star if Δ(B)≥3, and V (B) has at most three {u}-predecessors in L1(u) if Δ(B)≤2. Furthermore, if l=2 and B is a cycle, then |B|≡0 (mod 6).•A pentagraph is a simple graph without cycles on 3 or 4 vertices and without induced cycles of odd length at least 7.•If a non-bipartite pentagraph G contains a 5-hole C, then if C is not contained in any subgraph of G which is equal to the Petersen graph, then G[Li(V(C))] is bipartite for all i>0.•If a non-bipartite pentagraph G with a 5-hole contains an induced subgraph F which is equal to the Petersen graph, then G[Li(V(F))] is bipartite for all i>0.•For some special structure P, if a non-bipartite pentagraph G with a 5-hole contains P, then G[Li(V(P))] is bipartite for alli>0. A pentagraph is a simple graph without cycles on 3 or 4 vertices and without induced cycles of odd length at least 7. In this paper, we present some structural properties of pentagraphs. As a corollary, we give a short proof that there is exactly one 3-connected internally 4-connected non-bipartite cubic pentagraph, which is the Petersen graph.