Graphs and Combinatorics

, Volume 30, Issue 2, pp 395–410

Graphs Without Large Apples and the Maximum Weight Independent Set Problem

  • Vadim V. Lozin
  • Martin Milanič
  • Christopher Purcell
Original Paper

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


An appleAk is the graph obtained from a chordless cycle Ck 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 (Ak, Ak+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.


Claw-free graphsChordal graphsIndependent setpolynomial algorithm

Copyright information

© Springer Japan 2012

Authors and Affiliations

  • Vadim V. Lozin
    • 1
  • Martin Milanič
    • 2
    • 3
  • Christopher Purcell
    • 1
  1. 1.DIMAP and Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.University of Primorska, UP IAMKoperSlovenia
  3. 3.University of Primorska, UP FAMNITKoperSlovenia