, Volume 30, Issue 2, pp 395-410

Graphs Without Large Apples and the Maximum Weight Independent Set Problem

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Abstract

An apple A k is the graph obtained from a chordless cycle C k of length k ≥ 4 by adding a vertex that has exactly one neighbor on the cycle. The class of apple-free graphs is a common generalization of claw-free graphs and chordal graphs, two classes enjoying many attractive properties, including polynomial-time solvability of the maximum weight independent set problem. Recently, Brandstädt et al. showed that this property extends to the class of apple-free graphs. In the present paper, we study further generalization of this class called graphs without large apples: these are (A k , A k+1, . . .)-free graphs for values of k strictly greater than 4. The complexity of the maximum weight independent set problem is unknown even for k = 5. By exploring the structure of graphs without large apples, we discover a sufficient condition for claw-freeness of such graphs. We show that the condition is satisfied by bounded-degree and apex-minor-free graphs of sufficiently large tree-width. This implies an efficient solution to the maximum weight independent set problem for those graphs without large apples, which either have bounded vertex degree or exclude a fixed apex graph as a minor.

This paper extends and unifies some results that earlier appeared in proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms [26] and 33rd International Symposium on Mathematical Foundations of Computer Science [3]. V. V. Lozin gratefully acknowledges the support of DIMAP—the Centre for Discrete Mathematics and its Applications at the University of Warwick and from EPSRC, grant EP/I01795X/1. M. Milanič supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije”, research program P1–0285 and research projects J1–4010, J1–4021 and N1–0011.