Hitting all maximum stable sets in P₅-free graphs

We prove that every P₅-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where P_t denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H). More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest cardinality of a hitting set of all maximum stable sets in G, and ω(G) is the clique number of G. Also, C is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial. We introduce η-boundedness inspired by a question of Alon (asking how large η(G) can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely, • given a graph G, we have η(H)≤ω(H) for every induced subgraph H of G if and only if G is perfect; • there are graphs G with both η(G) and the girth of G arbitrarily large; and • if C is a hereditary class of graphs which is polynomially η-bounded, then C satisfies the Erdős-Hajnal conjecture. The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once. Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that P₅-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of P_t-free graphs. It remains open whether P_t-free graphs are η-bounded for t≥6. It also remains open whether P₅-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for P₅-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of P₅. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.