On the Structure of Bull-Free Perfect Graphs, 2: the Weakly Chordal Case

Figueiredo, Celina M. H. de; Maffray, Frédéric; Porto, Oscar

doi:10.1007/s003730170019

Cited by 4 publications

(13 citation statements)

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“…Theorem 3 [5]. For every bull-free weakly chordal graph, 1. if the graph has a P 6 , then it has a box partition, 2. if the graph has a box partition, then it has a homogeneous set or is transitively orientable.…”

Section: Box Partitions and Our Main Resultsmentioning

confidence: 99%

“…signi®cant progress on the conjecture, reducing it to the conjecture that a certain subclass of bull-free weakly chordal graphs is perfectly orderable [4,5]. Before explaining their results, we present some motivating background material.…”

Section: Overviewmentioning

confidence: 99%

“…Thus 2 holds. (1,2,3,4,5,6). In the upper (lower) matrix, the two vertices are adjacent (non-adjacent).…”

Section: Proofs Of Lemmas and Correctness Of Algorithmsmentioning

confidence: 99%

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Bull-Free Weakly Chordal Perfectly Orderable Graphs

Hayward¹

2001

Graphs Comb

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Using doubly lexical orders and the notion of box partition due to de Figueiredo, Maray, and Porto, we show that a certain subclass of bull-free weakly chordal graphs is perfectly orderable. This together with results of de Figueiredo, Maray, and Porto con®rms ChvaÂ tal's conjecture that bull-free graphs with no anti-hole and no odd hole are perfectly orderable; here hole means induced cycle with ®ve or more vertices.

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Section: Box Partitions and Our Main Resultsmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

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Bull-Free Weakly Chordal Perfectly Orderable Graphs

Hayward¹

2001

Graphs Comb

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“…If h ∈ P ∪ X ∪ Y and t ∈ T − Y − A this is by the definition of Y . Thus (7) holds. Next we claim that:…”

Section: Some Lemmasmentioning

confidence: 89%

“…So, by considering a neighbour a ′ of a in L 2 and a neighbour b ′ of b in L 2 , and by applying an analogous argument we obtain a path with the desired properties. Therefore (7) holds.…”

Section: Transitive Orientationsmentioning

confidence: 92%

Transitive orientations in bull-reducible Berge graphs

Figueiredo

Maffray

Maciel

2011

Discrete Applied Mathematics

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The perfection and recognition of bull-reducible Berge graphs

Everett

Figueiredo

Klein

et al. 2005

RAIRO-Theor. Inf. Appl.

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The recently announced Strong Perfect Graph Theorem states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph is bull-reducible if no vertex is in two bulls. In this paper we give a simple proof that every bull-reducible Berge graph is perfect. Although this result follows directly from the Strong Perfect Graph Theorem, our proof leads to a recognition algorithm for this new class of perfect graphs whose complexity, O(n 6), is much lower than that announced for perfect graphs.

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On the Structure of Bull-Free Perfect Graphs, 2: the Weakly Chordal Case

Cited by 4 publications

References 0 publications

Bull-Free Weakly Chordal Perfectly Orderable Graphs

Bull-Free Weakly Chordal Perfectly Orderable Graphs

Transitive orientations in bull-reducible Berge graphs

The perfection and recognition of bull-reducible Berge graphs

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