Graphs without two vertex-disjoint $S$-cycles

TL;DR

This paper investigates the hitting set size for $S$-cycles in graphs lacking two vertex-disjoint $S$-cycles, showing that four vertices suffice to hit all such cycles, extending Lovász's classical result.

Contribution

It establishes a bound of four vertices to hit all $S$-cycles in graphs with no two vertex-disjoint $S$-cycles, generalizing Lovász's theorem.

Findings

Provided a counterexample with 21 vertices where three vertices do not suffice.

Proved that four vertices are always enough to hit all $S$-cycles under the given conditions.

Abstract

Lov\'asz (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for $S$-cycles, when a graph has no two vertex-disjoint $S$-cycles. For a graph $G$ and a vertex set $S$ of $G$, an $S$-cycle is a cycle containing a vertex of $S$. We provide an example $G$ on $21$ vertices where $G$ has no two vertex-disjoint $S$-cycles, but three vertices are not sufficient to hit all $S$-cycles. On the other hand, we show that four vertices are enough to hit all $S$-cycles whenever a graph has no two vertex-disjoint $S$-cycles.