Algorithms for perfectly contractile graphs

We consider the class $\mathcal{A}$ of graphs that contain no odd hole, no antihole of length at least 5, and no “prism” (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class ${\cal A}'$ of graphs that contain no odd hole, no antihole of length at least 5 and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. We proved previously that every graph $G\in \mathcal{A}$ is “perfectly contractile”, as conjectured by Everett and Reed [see the chapter “Even pairs” in the book {\it Perfect Graphs}, J.L. Ram'{\i}rez-Alfons'{\i}n and B.A. Reed, eds., Wiley Interscience, 2001]. The analogous conjecture concerning graphs in ${\mathcal{A}}^{\mathrm{\prime }}$ is still open.