Describing hereditary properties by forbidden circular orderings

Santiago Guzmán-Pro, Pavol Hell and César Hernández-Cruz

Applied Mathematics and Computation, 2023, vol. 438, issue C

Abstract: Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is k-colourable if and only if V(G) admits a linear ordering ≤ with no vertices v1≤⋯≤vk+1 such that vivi+1∈E(G) for every i∈{1,⋯,k}. In this paper, we study such characterizations when the extra structure is a circular ordering of the vertex set. We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than k. In fact, every description by finitely many forbidden circularly ordered graphs can be translated to a description by finitely many forbidden linearly ordered graphs. Nevertheless, our observations underscore the fact that in many cases the circular order descriptions are nicer and more natural.

Keywords: Graph classes; Vertex orderings with forbidden patterns; Forbidden subgraph characterizations (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006294

DOI: 10.1016/j.amc.2022.127555

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