Path Spectra and Forbidden Families

The path spectrum, \(\operatorname{sp}(G)\), of a graph \(G\) is the set of all lengths of maximal paths in \(G\). The path spectrum is continuous if \(\operatorname{sp}(G) = \{\ell, \ell1, \dots, \ell\}\) for some \(\ell \leq m\). A graph whose path spectrum consists of a single element is called scent and is by definition continuous. In this paper, we determine when a \(\{K_{1, 3}, S\}\)-free graph has a continuous path spectrum where \(S\) is one of \(C_3, P_4, P_5, P_6, Z_1, Z_2, Z_3, N, B\), or \(W\).