The Price of Connectivity for Cycle Transversals

Abstract

For a family of graphs ${\mathcal F}$, an $\mathcal {F}$-transversal of a graph G is a subset $S \subseteq V(G)$ that intersects every subset of V(G) that induces a subgraph isomorphic to a graph in $\mathcal {F}$. Let $t_\mathcal {F}(G)$ be the minimum size of an $\mathcal {F}$-transversal of G, and $ct _\mathcal {F}(G)$ be the minimum size of an $\mathcal {F}$-transversal of G that induces a connected graph. For a class of connected graphs $\mathcal{G}$, the price of connectivity for $\mathcal {F}$-transversals is the supremum of the ratios $ct _\mathcal {F}(G)/t_\mathcal {F}(G)$ over all $G\in \mathcal{G}$. We perform an in-depth study into the price of connectivity for various well-known graph families $\mathcal{F}$ that contain an infinite number of cycles and that, in addition, may contain one or more anticycles or short paths. For each of these families we study the price of connectivity for classes of graphs characterized by one forbidden induced subgraph H. We determine exactly those classes of H-free graphs for which this graph parameter is bounded by a multiplicative constant, bounded by an additive constant, or equal to 1. In particular, our tetrachotomies extend known results of Belmonte et al. (EuroComb 2012, MFCS 2013) for the case when $\mathcal{F}$ is the family of all cycles.

This work was supported by a London Mathematical Society Scheme 4 Grant and by EPSRC Grant EP/K025090/1, and, in part, by the Slovenian Research Agency (I0-0035, research program P1-0285, research projects N1-0032, J1-5433, J1-6720, and J1-6743, and a Young Researchers Grant).