Erdos-Pósa property of obstructions to interval graphs

The duality between packing and covering problems lies at the heart of fundamental combinatorial proofs, as well as well-known algorithmic methods such as the primal-dual method for approximation and win/win-approach for parameterized analysis. The very essence of this duality is encompassed by a well-known property called the Erdos-Pósa property, which has been extensively studied for over five decades. Informally, we say that a class of graphs F admits the Erdos-Pósa property if there exists f such that for any graph G, either G has k vertex-disjoint “copies” of the graphs in F, or there is a set S ? V (G) of f(k) vertices that intersects all copies of the graphs in F. In the context of any graph class G, the most natural question that arises in this regard is as follows - do obstructions to G have the Erdos-Pósa property? Having this view in mind, we focus on the class of interval graphs. Structural properties of interval graphs are intensively studied, also as they lead to the design of polynomial-time algorithms for classic problems that are NP-hard on general graphs. Nevertheless, about one of the most basic properties of such graphs, namely, the Erdos-Pósa property, nothing is known. In this paper, we settle this anomaly: We prove that the family of obstructions to interval graphs - namely, the family of chordless cycles and ATs - admits the Erdos-Pósa property. Our main theorem immediately results in an algorithm to decide whether an input graph G has k vertex-disjoint ATs and chordless cycles, or there exists a set of O(k² log k) vertices in G that hits all ATs and chordless cycles.