Graphs Without Large Apples and the Maximum Weight Independent Set Problem Authors Vadim V. Lozin DIMAP and Mathematics Institute University of Warwick Martin Milanič University of Primorska, UP IAM University of Primorska, UP FAMNIT Christopher Purcell DIMAP and Mathematics Institute University of Warwick Original Paper

First Online: 15 November 2012 Received: 25 May 2009 Revised: 18 October 2012 DOI :
10.1007/s00373-012-1263-y

Cite this article as: Lozin, V.V., Milanič, M. & Purcell, C. Graphs and Combinatorics (2014) 30: 395. doi:10.1007/s00373-012-1263-y
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.

Keywords Claw-free graphs Chordal graphs Independent set polynomial algorithm 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.

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