Avoidable Paths in Graphs

Bonamy, Marthe; Defrain, Oscar; Hatzel, Meike; Thiebaut, Jocelyn

doi:10.37236/9030

Cited by 7 publications

(11 citation statements)

References 2 publications

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“…Regarding avoidable paths on k vertices, one needs to detect first with a naive algorithm a path P k in O(n k ) time and then test whether P k being avoidable or not. As observed in [9], such a detection is nearly optimal, since we can hardly avoid the dependence of the exponent in O(n k ). Therefore by Theorem 25 we get an O(n k+1 • m)-algorithm for listing all avoidable paths on k vertices.…”

Section: Discussionmentioning

confidence: 96%

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Computing and Listing Avoidable Vertices and Paths

Papadopoulos¹,

Zisis²

2021

Preprint

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A simplicial vertex of a graph is a vertex whose neighborhood is a clique. It is known that listing all simplicial vertices can be done in O(nm) time or O(n ω ) time, where O(n ω ) is the time needed to perform a fast matrix multiplication. The notion of avoidable vertices generalizes the concept of simplicial vertices in the following way: a vertex u is avoidable if every induced path on three vertices with middle vertex u is contained in an induced cycle. We present algorithms for listing all avoidable vertices of a graph through the notion of minimal triangulations and common neighborhood detection. In particular we give algorithms with running times O(n 2 m) and O(n 1+ω ), respectively. However, we propose a faster algorithm that runs in time O(n 2 + m 2 ), and thus matches the corresponding running time of listing the simplicial vertices on sparse graphs with m = O(n). To complement our results, we consider their natural generalizations of avoidable edges and avoidable paths. We propose an O(nm)-time algorithm that recognizes whether a given induced path is avoidable.

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Section: Discussionmentioning

confidence: 96%

“…al [3] proved first that every non-edgeless graph contains an avoidable edge, considering the case of k = 2. Regarding the existence of an avoidable induced path of arbitrary length, Bonamy et al [9] settled a conjecture in [3] and showed that every graph is either P k -free or contains an avoidable P k .…”

Section: Introductionmentioning

confidence: 99%

Computing and Listing Avoidable Vertices and Paths

Papadopoulos¹,

Zisis²

2021

Preprint

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“…We prove Theorem 8 by adapting the approach used by Bonamy et al [4] to prove that for every positive integer

k

, every graph that contains an induced

P_{k}

also contains an avoidable induced

P_{k}

(case ind of Corollary 5).…”

Section: Induced Pathsmentioning

confidence: 99%

“…We adapt the approach of [4] to shifting. For a graph

G

and a positive integer

k

, we say that: property

H_{B} (G, k)

holds if every induced path

P_{k}

G

can be shifted to an avoidable induced path; for a vertex

v \in V (G)

, property

H_{R} (G, k, v)

holds if every induced path

P_{k}

G - N [v]

can be shifted in

G - N [v]

to an avoidable induced path in

G

; property

H_{R} (G, k)

holds if for every

v \in V (G)

we have

H_{R} (G, k, v)

.…”

Section: Induced Pathsmentioning

confidence: 99%

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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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Graphs with at most two moplexes

Dallard,

Ganian,

Hatzel

et al. 2024

Journal of Graph Theory

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A moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw‐free, and diamond‐free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of ‐moplex graphs, which are defined as graphs containing at most moplexes. Of particular interest is the smallest nontrivial case , which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2‐moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity‐theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max‐Cut. On the other hand, we prove that every connected 2‐moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

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Avoidable Paths in Graphs

Cited by 7 publications

References 2 publications

Computing and Listing Avoidable Vertices and Paths

Computing and Listing Avoidable Vertices and Paths

Shifting paths to avoidable ones

Graphs with at most two moplexes

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