Graphs with all holes the same length (Q6564607)

A graph is called \(\ell\)-holed if all its induced cycles of length at least four have length exactly \(\ell\). In this paper, the authors give a complete description of the \(\ell\)-holed graphs for each \(\ell\geq 7\) as follows: Let \(G\) be a graph with no clique cutset and no universal vertex (a vertex of \(G\) adjacent to all the other vertices of \(G\)), and let \(\ell\geq 7\). Then \(G\) is \(\ell\)-holed if and only if either \(G\) is a blow-up of a cycle of length \(\ell\), or \(G\) is a blow-up of an \(\ell\)-framework, where the terms blow-up of an \(\ell\)-cycle and \(\ell\)-framework are defined in this paper.