Cameron, K, Da Silva, MVG, Huang, S et al. (1 more author) (2018) Structure and algorithms for (cap, even hole)-free graphs. Discrete Mathematics, 341 (2). pp. 463-473. ISSN 0012-365X
Abstract
A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most [View the MathML source], and hence [View the MathML source], where ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set, where n is the number of vertices and m is the number of edges of the input graph. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G)−1 and clique-width at most 48.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 Elsevier B.V. This is an author produced version of a paper published in Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Even-hole-free graph; Structure theorem; Decomposition; Combinatorial optimization; Coloring; Maximum weight stable set; Treewidth; Clique-width |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Funding Information: | Funder Grant number EPSRC EP/N019660/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 18 Jul 2017 10:12 |
Last Modified: | 13 Oct 2018 00:38 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.disc.2017.09.013 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:119116 |