Graphs Without Large Apples and the Maximum Weight Independent Set Problem

• Published in: Graphs and combinatorics Vol. 30; no. 2; pp. 395 - 410
• Main Authors: Lozin, Vadim V., Milanič, Martin, Purcell, Christopher
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.03.2014; Springer Nature B.V
• Subjects: Apexes; Apples; Combinatorial analysis; Combinatorics; Complexity; Engineering Design; Graphs; Mathematical problems; Mathematics; Mathematics and Statistics; Original Paper; Polynomials; Claw-free graphs; Independent set; polynomial algorithm; Chordal graphs
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-012-1263-y

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.