Vuskovic, K and Horsfield, J orcid.org/0000-0002-4388-5123 (2021) Two classes of β-perfect graphs that do not necessarily have simplicial extremes. Discrete Mathematics, 344 (7). 112374. ISSN 0012-365X
Abstract
For a graph G, β(G) = max{δ(G′)+1 | G′ is an induced subgraph of G}. This parameter is an upper bound on the chromatic number of a graph. A graph G is β-perfect if χ(G′) = β(G′) for all induced subgraphs G′ of G. A number of classes have been shown in literature to be β-perfect, but for the class of all β-perfect graphs, the complexity of their recognition and characterization in terms of forbidden induced subgraphs remain open.
It is known that minimally β-imperfect graphs cannot have a simplicial extreme, i.e. a vertex whose neighborhood is a clique or of size 2. β-perfection of all the known β-perfect classes of graphs was shown through the existence of simplicial extremes. In this paper we study two classes of β-perfect graphs that do not necessarily have this property.
A hole is a chordless cycle of length at least 4, and it is even or odd depending on the parity of its length. β-perfect graphs cannot contain even holes as induced subgraphs. We prove that graphs that do not contain an even hole, a twin wheel nor a cap as an induced subgraph are β-perfect (where a twin wheel is a graph that consists of a hole and a vertex that has exactly 3 neighbors on the hole, that are furthermore consecutive on the hole; and a cap is a graph that consist of a hole and a vertex that has exactly 2 neighbors on the hole, that are furthermore consecutive on the hole). This class properly contains chordal graphs, and is the only known generalization of chordal graphs that is shown to be β-perfect.
A hyperhole is a graph that is obtained from a hole by a sequence of clique substitutions. We give a complete structural characterization of β-perfect hyperholes, which we then use to give a linear-time algorithm to recognize whether a hyperhole is β-perfect. We also obtain a complete list of minimally β-imperfect hyperholes.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | Crown Copyright © 2021 Published by Elsevier B.V. All rights reserved. This is an author produced version of an article published in Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | B-perfect graph; Even-hole-free graph; Greedy coloring algorithm; No simplicial extreme |
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 (Engineering and Physical Sciences Research Council) EP/N019660/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 12 Feb 2021 12:03 |
Last Modified: | 31 Mar 2022 00:38 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.disc.2021.112374 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:171046 |