1-perfectly orientable K 4 -minor-free and outerplanar graphs

Brešar, Boštjan; Kos, Tim; Hartinger, Tatiana Romina; Milanič, Martin

doi:10.1016/j.endm.2016.09.035

Cited by 6 publications

(9 citation statements)

References 20 publications

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“…One of the results from the table, namely the (tw, ω)-boundedness of the class of K 2,3 -induced-minor-free graphs, implies that the class of 1-perfectly orientable graphs is (tw, ω)-bounded. This answers a question raised by Brešar et al [10].…”

Section: Our Resultssupporting

confidence: 67%

“…As explained by Belmonte et al [3] (and observed also in [10]), the following fact can be derived from the proof of Theorem 9 in [38].…”

Section: Back To Treewidthmentioning

confidence: 79%

“…A graph G is said to be 1-perfectly orientable if it has an orientation D such that for every vertex v ∈ V (G), the out-neighborhood of v in D is a clique in G. The class of 1-perfectly orientable graphs is a common generalization of the classes of chordal graphs and circular-arc graphs. While 1-perfectly orientable graphs were studied in several papers (see, e.g., [2,10,34]), their structure remains poorly understood. Brešar et al showed in [10] that the treewidth of every 1-perfectly orientable planar graph is at most 21 and asked whether the class of 1-perfectly orientable graphs is (tw, ω)-bounded.…”

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confidence: 99%

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Treewidth versus clique number. I. Graph classes with a forbidden structure

Dallard,

Milanič,

Štorgel

2020

Preprint

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Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes in which this condition is also sufficient, which we call (tw, ω)-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and k-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor, and induced subgraph relations. For each of them, we give a complete characterization of the graphs H for which the class of graphs excluding H is (tw, ω)-bounded. Our results imply that the class of 1-perfectly orientable graphs is (tw, ω)-bounded, answering a question of Brešar, Hartinger, Kos, and Milanič from 2018. We also reveal some further algorithmic implications of (tw, ω)-boundedness related to list k-coloring and clique problems.

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Section: Our Resultssupporting

confidence: 67%

“…As explained by Belmonte et al [3] (and observed also in [10]), the following fact can be derived from the proof of Theorem 9 in [38].…”

Section: Back To Treewidthmentioning

confidence: 79%

mentioning

confidence: 99%

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Treewidth versus clique number. I. Graph classes with a forbidden structure

Dallard,

Milanič,

Štorgel

2020

Preprint

Self Cite

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“…A graph G is said to be 1-perfectly orientable if its edges can be oriented so that no vertex has a pair of non-adjacent out-neighbors. The class of 1-perfectly orientable graphs was introduced in 1982 by Skrien [108] and studied by Bang-Jensen et al [6] and more recently by Hartinger and Milanič [80] and Brešar et al [28]. While the class of 1-perfectly orientable graphs is known to be a common generalization of the classes of chordal graphs and circular-arc graphs, the structure of graphs in this class remains elusive.…”

Section: Reducing the Problem To Triconnected Componentsmentioning

confidence: 99%

Treewidth versus clique number. II. Tree-independence number

Dallard¹,

Milanič²,

Štorgel³

2021

Preprint

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In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw, ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether (tw, ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent H-Packing problem, for any fixed finite set H of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw, ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of...

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“…Studying the structure of B 1 -free orientable graphs has caught the interest of several authors. In particular, Hartinger and Milanic, and the same authors with Brešar and Kos, have thoroughly studied this family in a series of papers [5,7,8]. We will follow their terminology and call the class of {B 1 }-graphs, 1-perfectly-orientable graphs (1-p.o.…”

Section: Introductionmentioning

confidence: 99%

Orientations without forbidden patterns on three vertices

Guzmán-Pro¹,

Hernández-Cruz²

2020

Preprint

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Given a set of oriented graphs F , a graph G is an F -graph if it admits an F -free orientation. Building on previous work by Bang-Jensen and Urrutia, we propose a master algorithm that determines if a graph admits an F -free orientation when F is a subset of the orientations of P 3 and the transitive triangle.We extend previous results of Skrien by studying the class of F -graphs, when F is any set of oriented graphs of order three; structural characterizations for all such sets are provided, except for the so-called perfectly-orientable graphs and one of its subclasses, which remain as open problems.

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1-perfectly orientable K 4 -minor-free and outerplanar graphs

Cited by 6 publications

References 20 publications

Treewidth versus clique number. I. Graph classes with a forbidden structure

Treewidth versus clique number. I. Graph classes with a forbidden structure

Treewidth versus clique number. II. Tree-independence number

Orientations without forbidden patterns on three vertices

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