Hitting all maximum stable sets in P5-free graphs

We prove that every $P_5$-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 $h:N\to N$ such that $\eta \left(G\right)\le h\left(\omega \right(G\left)\right)$ for every graph $G\in C$, where $\eta \left(G\right)$ denotes smallest cardinality of a hitting set of all maximum stable sets in G, and $\omega \left(G\right)$ 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 $\eta \left(G\right)$ 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 $\eta \left(H\right)\le \omega \left(H\right)$ for every induced subgraph H of G if and only if G is perfect;

• •

  there are graphs G with both $\eta \left(G\right)$ 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 $P_5$-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 $t\ge 6$.

It also remains open whether $P_5$-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for $P_5$-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of $P_5$. 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.