Graph Classes and Forbidden Patterns on Three Vertices

Feuilloley, Laurent; Habib, Michel

doi:10.1137/19m1280399

Cited by 14 publications

(43 citation statements)

References 51 publications

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“…In particular, since we already showed that outerplanar graphs can be naturally described by forbidden circularly ordered graphs (Proposition 7) by Observation 8, we recover an observation mentioned in [4] that states that there is a finite set of linearly ordered patterns that characterizes outerplanar graphs. The class of circular-arc graphs is also described by finitely many forbidden circularly ordered graphs (Proposition 6) so there is a set of linearly ordered patterns on four vertices F C such that the class of graphs that admit an F Cfree linear ordering is the class of circular-arc graphs.…”

Section: Circular Arrangements and Linearly Ordered Patternssupporting

confidence: 75%

“…To use a technique analogous to the one used in [4] for depicting families of linearly ordered graphs, we introduce circularly ordered patterns. A pattern consists of a set V together with a set of edges E and a set of non-edges N E with the restriction that…”

Section: Circularly Ordered Graphsmentioning

confidence: 99%

“…As noted by Habib and Feuilloley [4], an obvious line of research in the context of forbidden linearly ordered graphs, is to study hereditary properties characterized by forbidden sets of linearly ordered graphs on four vertices or more. To this end, we notice that for any hereditary property described by a finite set of forbidden circularly ordered graphs, there is a set of linearly ordered graphs (with the same size of vertex sets) that describes the same property.…”

Section: Circular Arrangements and Linearly Ordered Patternsmentioning

confidence: 99%

“…Linear forests, caterpillar forests and forests are examples of graph classes characterized by a set of forbidden linearly ordered patterns on three vertices [4].…”

Section: Circular Arrangements and Linearly Ordered Patternsmentioning

confidence: 99%

“…Around 2014, Hell, Mohar and Rafiey [7] showed that for every set F of linearly ordered graphs on three vertices, the class of graphs that admit an Ffree linear ordering can be recognized in polynomial time. Recently, Habib and Feuilloley published a thorough survey [4] on the subject, where they characterized all hereditary properties defined by forbidden linear ordering on three vertices. Moreover, they showed that all of these classes (except for two of them) can be recognized in linear time.…”

Section: Introductionmentioning

confidence: 99%

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Describing hereditary properties by forbidden circular orderings

Guzmán-Pro¹,

Hell²,

Hernández-Cruz³

2021

Preprint

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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 ≤ • • • ≤ v k+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.

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Section: Circular Arrangements and Linearly Ordered Patternssupporting

confidence: 75%

Section: Circularly Ordered Graphsmentioning

confidence: 99%

Section: Circular Arrangements and Linearly Ordered Patternsmentioning

confidence: 99%

“…Linear forests, caterpillar forests and forests are examples of graph classes characterized by a set of forbidden linearly ordered patterns on three vertices [4].…”

Section: Circular Arrangements and Linearly Ordered Patternsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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