Partial Characterizations of 1‐Perfectly Orientable Graphs

Hartinger, Tatiana Romina; Milanič, Martin

doi:10.1002/jgt.22067

Cited by 14 publications

(21 citation statements)

References 31 publications

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“…Our algorithm for the maximum weight clique problem in the class of hole-cyclically orientable graphs will be based on the fact that the classes of 1-perfectly orientable and hole-cyclically orientable graphs coincide within the class of cobipartite graphs, where they also coincide with circular-arc graphs. The equivalence between properties 1, 3 and 4 in the lemma below was already observed in [27]. Due to Lemma 5.1, the list can be trivially extended with the hole-cyclically orientable property.…”

Section: Implications For the Maximum Weight Clique Problemmentioning

confidence: 64%

“…Moreover, we do this even in the more general setting of hole-cyclically orientable graphs. The fact that every 1-perfectly orientable graph is hole-cyclically orientable is a consequence of the following simple lemma (see, e.g., [27]). On the other hand, not every hole-cyclically orientable graph is 1-perfectly orientable, as can be seen from Figure 3.…”

Section: Implications For the Maximum Weight Clique Problemmentioning

confidence: 99%

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Avoidable Vertices and Edges in Graphs

Beisegel

Chudnovsky

Gurvich

et al. 2019

Lecture Notes in Computer Science

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A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF -vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle. * An extended abstract of this work is

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Section: Implications For the Maximum Weight Clique Problemmentioning

confidence: 64%

Section: Implications For the Maximum Weight Clique Problemmentioning

confidence: 99%

Avoidable Vertices and Edges in Graphs

Beisegel

Chudnovsky

Gurvich

et al. 2019

Lecture Notes in Computer Science

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“…The class of 1-perfectly orientable graphs is closed under induced minors [18]. The characterization of the class of 1-perfectly orientable graphs in terms of forbidden induced minors is not known; a partial answer is given in the following theorem.…”

Section: Minors and Induced Minorsmentioning

confidence: 99%

“…The subclass of 1-perfectly orientable graphs consisting of graphs that admit an orientation that is both an in-tournament and an out-tournament was characterized in [25] (see also [19]) as precisely the class of proper circular arc graphs. A characterization of 1-perfectly orientable graphs in terms of edge clique covers and characterizations of 1-perfectly orientable co-bipartite graphs and cographs were obtained recently by Hartinger and Milanič [18].…”

Section: Introductionmentioning

confidence: 99%

“…Organization of the paper. In Section 2 we fix the notation and state some preliminary results for later use; in particular, a result from [18] gives a long list of forbidden induced minors for the class of 1-perfectly orientable graphs. In this section we also prove some preparatory results for later use; for instance, cyclically orientable graphs are characterized as exactly the {K 4 , K 2,3 }-inducedminor-free graphs and it is proved that every 1-perfect orientation of a connected graph contains at most one sink.…”

Section: Introductionmentioning

confidence: 99%

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

Brešar

Kos

Hartinger

et al. 2016

Electronic Notes in Discrete Mathematics

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A graph G is said to be 1-perfectly orientable if it has an orientation such that for every vertex v ∈ V (G), the out-neighborhood of v in D is a clique in G. In 1982, Skrien posed the problem of characterizing the class of 1-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to 2-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of 1-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of 1-perfectly orientable K 4 -minor-free graphs and of 1-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of 2-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most 2. *

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Bisimplicial separators

Milanič,

Penev,

Pivač

et al. 2024

Journal of Graph Theory

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A minimal separator of a graph is a set such that there exist vertices with the property that separates from in , but no proper subset of does. For an integer , we say that a minimal separator is ‐simplicial if it can be covered by cliques and denote by the class of all graphs in which each minimal separator is ‐simplicial. We show that for each , the class is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for . We also give a complete list of minimal forbidden induced minors for . Next, we show that, for , every nonnull graph in has a ‐simplicial vertex, that is, a vertex whose neighborhood is a union of cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in . Further, we show that, for , it is NP‐hard to recognize graphs in ; the time complexity of recognizing graphs in is unknown. We also show that the Maximum Clique problem is NP‐hard for graphs in . Finally, we prove a decomposition theorem for diamond‐free graphs in (where the diamond is the graph obtained from by deleting one edge), and we use this theorem to obtain polynomial‐time algorithms for the Vertex Coloring and recognition problems for diamond‐free graphs in , and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

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Partial Characterizations of 1‐Perfectly Orientable Graphs

Cited by 14 publications

References 31 publications

Avoidable Vertices and Edges in Graphs

Avoidable Vertices and Edges in Graphs

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

Bisimplicial separators

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