Let $G$ be a graph without any hole or antihole of odd length at least 5 (i.e. $G$ is a Berge graph and so by the Strong Perfect Graph Theorem, $G$ is perfect). Assume further that $G$ has no antihole of length 6 and also it has no even hole or antihole of length at least 8. Also suppose that $G$ has no induced path $P_6$ of length 6.
Thus graphs in question are perfect and $P_6$-free with a slightly stronger property that they have no antihole of length 6.
Is there a classification of such graphs?
Of course one may expect that by docomposion results due to [M. Chudnovsky, N. Robertson, P.D. Seymour, R.Thomas, V. Chvátal, N. Sbihi, M. Conforti, G. Cornuéjols, and K. Vušković] for perfect graphs and also by the results due to [R. Mosca, Pim van 't Hof and DaniÄel Paulusma] on P6-free graph, it may be possible to find a characterization or even a classification; but, for a lazy man as me, it is better to know first if such a characterization has been aleardy seen or used.
The motivation is this question of mine about graphs $G$ which are cospectral with a friendship graph. Such a graph $G$ has no induced subgraph with two eigenvalues greater than 1 and also no induced subgraph of $G$ has two eigenvalues less than -1. The latter follows from the interlacing lemma.
Thus it is easily seen that such graphs $G$ are perfect and $P_6$-free and they have no antihole of length 6.
