Avoidable Vertices and Edges in Graphs

Beisegel, Jesse; Chudnovsky, Maria; Gurvich, Vladimir; Milanič, Martin; Servatius, Mary

doi:10.1007/978-3-030-24766-9_10

Cited by 9 publications

(29 citation statements)

References 40 publications

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“…This result complements the O(n(n log n + m log log n)) algorithm for the Maximum Weight Clique problem on 1-perfectly orientable graphs with n vertices and m edges due to Beisegel et al [7] and a linear-time algorithm for the List k-Coloring problem in the same class of graphs, which follows from the corresponding algorithm for (tw, ω)-bounded graph classes due to Chaplick and Zeman [38] and the fact that the class of K 2,3 -induced-minor-free graphs is (tw, ω)-bounded (by a computable binding function) [54,55].…”

Section: Reducing the Problem To Triconnected Componentssupporting

confidence: 76%

“…Additionally, the fact that MWIS is solvable in polynomial time in the class of K 2,3 -induced-minor-free graphs implies a polynomial-time solution for MWIS in the class of 1-perfectly orientable graphs, which is a common generalization of the classes of chordal graphs and circular-arc graphs. This answers a question of Beisegel et al posed in [7]. The analogous question for k-coloring was answered in [54,55].…”

Section: Related Work On the Mwis Problemsupporting

confidence: 56%

“…For every q ≥ 2, the MWIS problem is solvable in time O(n 2q ) on n-vertex K 2,qinduced-minor-free graphs. Corollary 9.12 answers the question regarding the complexity of the Maximum Independent Set problem in the class of 1-perfectly orientable graphs posed by Beisegel et al in [7]. 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.…”

Section: Reducing the Problem To Triconnected Componentsmentioning

confidence: 83%

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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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Section: Reducing the Problem To Triconnected Componentssupporting

confidence: 76%

Section: Related Work On the Mwis Problemsupporting

confidence: 56%

Section: Reducing the Problem To Triconnected Componentsmentioning

confidence: 83%

See 1 more Smart Citation

Treewidth versus clique number. II. Tree-independence number

Dallard¹,

Milanič²,

Štorgel³

2021

Preprint

Self Cite

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show abstract

“…In Corollary 6, the case i = 3 is the most important case. The corresponding statement was conjectured by Beisegel et al [2] and proved for ℓ = 1. For ℓ = 0 the result follows from earlier works [8,9].…”

Section: Related Workmentioning

confidence: 57%

“…For ℓ = 0 the result follows from earlier works [8,9]. We refer to [2] for motivation and more details. Bonamy et al [4] recently proved the conjecture for all ℓ.…”

Section: Related Workmentioning

confidence: 68%

Shifting paths to avoidable ones

Gurvich¹,

Krnc²,

Milanič³

et al. 2020

Preprint

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An extension of an induced path P in a graph G is an induced path P ′ such that deleting the endpoints of P ′ results in P . An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. Beisegel, Chudovsky, Gurvich, Milani, and Servatius (WADS 2019) conjectured that every graph that contains an induced k-vertex path also contains an avoidable induced path of the same length, and proved the result for k = 2 (the case k = 1 was known before). The same year Bonamy, Defrain, Hatzel, and Thiebaut proved the conjecture for all k. Here we strengthen their result and show that every induced path can be shifted to an avoidable one. We also prove analogous results for the non-induced case and for the case of walks, and show that the statement cannot be extended to trails or to isometric paths.

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Shifting paths to avoidable ones

Gurvich

Krnc

Milanič

et al. 2021

Journal of Graph Theory

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An extension of an induced path P in a graph G is an induced path P false′ such that deleting the endpoints of P false′ results in P. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced k‐vertex path also contains an avoidable induced path of the same length, and proved the result for k = 2. The case k = 1 was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all k in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced k‐vertex paths are shifts of each other if their union is an induced path with k + 1 vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.

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Avoidable Vertices and Edges in Graphs

Cited by 9 publications

References 40 publications

Treewidth versus clique number. II. Tree-independence number

Treewidth versus clique number. II. Tree-independence number

Shifting paths to avoidable ones

Shifting paths to avoidable ones

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