Partial Characterizations of Circular-Arc Graphs

Bonomo, Flavia; Durán, Guillermo; Grippo, Luciano N.; Safe, Martín D.

doi:10.1016/j.endm.2008.01.009

Cited by 4 publications

(12 citation statements)

References 15 publications

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“…Finding a forbidden subgraph characterization of circular-arc graphs is a challenging open problem studied since the late 1960s [14,21,22,23,24]. Many partial results toward this goal have been proposed over the years, but a full answer remains elusive, capturing the interest of many researchers [1,2,6,14,17,21,22,23,24]. A certifying algorithm is an algorithm that outputs a certificate, along with its answer (be it positive or negative), where the certificate can be used to easily justify the given answer.…”

Section: Introductionmentioning

confidence: 99%

Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm

Francis

Hell

Stacho

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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A circular-arc graph is the intersection graph of arcs of a circle. It is a well-studied graph model with numerous natural applications. A certifying algorithm is an algorithm that outputs a certificate, along with its answer (be it positive or negative), where the certificate can be used to easily justify the given answer. While the recognition of circular-arc graphs has been known to be polynomial since the 1980s, no polynomial-time certifying recognition algorithm is known to date, despite such algorithms being found for many subclasses of circular-arc graphs. This is largely due to the fact that a forbidden structure characterization of circular-arc graphs is not known, even though the problem has been intensely studied since the seminal work of Klee in the 1960s.In this contribution, we settle this problem. We present the first forbidden structure characterization of circular-arc graphs. Our obstruction has the form of mutually avoiding walks in the graph. It naturally extends a similar obstruction that characterizes interval graphs. As a consequence, we give the first polynomial-time certifying algorithm for the recognition of circular-arc graphs.

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Section: Introductionmentioning

confidence: 99%

Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm

Francis

Hell

Stacho

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In [5], circular-arc graphs are characterised within some graph classes including, among others, the class of diamond-free graphs. The following is a straightforward corollary of Theorem 16 in [5].…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Circular-arc graphs can be recognised in linear time [29], and have been characterised recently by a family of obstacles [18]. Previously, partial characterisations by minimal forbidden induced subgraphs were presented in [32] and [5].…”

Section: Basic Definitionsmentioning

confidence: 99%

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Characterising circular-arc contact B0–VPG graphs

Bonomo

Galby

Gonzalez

2020

Discrete Applied Mathematics

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A contact B 0 -VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-toone correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown in [15] that Recognition is NP-complete for contact B 0 -VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B 0 -VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

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“…On the one hand, as observed in [35], if H is isomorphic to claw, 4-wheel, 5-wheel, or tent, then Q H ( ) does not have the circular-ones property for rows. On the other hand, if H is isomorphic to net, C 6 , or C* k for some ≥ k 4, then it is also the case that Q H ( ) does not have the circular-ones property for rows because each of Q (net) and Q C ( ) 6 contains M IV as a configuration and Q C ( ) * k contains M k ( ) * I as a configuration for each ≥ k 4. Since H is an induced subgraph of G Q H , ( ) is contained in Q G ( ) as a configuration.…”

Section: Corollary 42 a Graph G Is A Proper Helly Circular-arc Graphmentioning

confidence: 99%

Characterization and linear‐time detection of minimal obstructions to concave‐round graphs and the circular‐ones property

Safe

2019

Journal of Graph Theory

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A graph is concave‐round if its vertices can be circularly enumerated so that the closed neighborhood of each vertex is an interval in the enumeration. In this study, we give a minimal forbidden induced subgraph characterization for the class of concave‐round graphs, solving a problem posed by Bang‐Jensen, Huang, and Yeo [SIAM J. Discrete Math., 13 (2000), pp. 179–193]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any given graph that is not concave‐round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices for the circular‐ones property for rows and for the circular‐ones property for rows and columns and show that, also for both variants of the property, one of the corresponding forbidden submatrices can be found (if present) in any given matrix in linear time. We make some final remarks regarding connections to some classes of circular‐arc graphs.

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Partial Characterizations of Circular-Arc Graphs

Cited by 4 publications

References 15 publications

Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm

Forbidden structure characterization of circular-arc graphs and a certifying recognition algorithm

Characterising circular-arc contact B0–VPG graphs

Characterization and linear‐time detection of minimal obstructions to concave‐round graphs and the circular‐ones property

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