Hitting all maximum stable sets in $P_5$-free graphs
Sepehr Hajebi, Yanjia Li, Sophie Spirkl
Abstract
We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets. More generally, let us say a class $\mathcal{C}$ of graphs is $η$-bounded if there exists a function $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $η(G)\leq h(ω(G))$ for every graph $G\in \mathcal{C}$, 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, $\mathcal{C}$ 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 and motivated by a number of meaningful similarities to $χ$-boundedness. In particular, we propose 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 $t\geq 6$. 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 $P_5$.