Intersection of longest paths in graph classes

Cerioli, Márcia R.; Lima, Paloma T.

doi:10.1016/j.dam.2019.03.022

Cited by 19 publications

(12 citation statements)

References 15 publications

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“…Transversals of longest paths has been also studied [4,5,6,8,13,24,25]. Also, other questions about intersection of longest cycles have been rised by several authors [7,20,22,27].…”

Section: Discussionmentioning

confidence: 99%

Transversals of Longest Cycles in Partial $k$-Trees and Chordal Graphs

Gutiérrez¹

2019

Preprint

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“…Transversals of longest paths has been also studied [4,5,6,8,13,24,25]. Also, other questions about intersection of longest cycles have been rised by several authors [7,20,22,27].…”

Section: Discussionmentioning

confidence: 99%

Transversals of Longest Cycles in Partial $k$-Trees and Chordal Graphs

Gutiérrez¹

2019

Preprint

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“…for every connected graph G on n vertices, and lpt(G) ≤ k + 1 for every connected partial k-tree G. The latter result implies that lpt(H) ≤ ω(H), where H is a connected chordal graph and ω(H) is the clique number of H. Furthermore, Cerioli et al [4] showed that lpt(H) ≤ max{1, ω(H) − 2}. Recently, Harvey and Payne [11] improved this result and proved that lpt(H) ≤ 4 • ω(H)…”

mentioning

confidence: 90%

“…The answer to Gallai's question is affirmative for some special classes of graphs such as split graphs [14], circular-arc graphs [1,13], outerplanar graphs and 2-trees [17], series-parallel graphs [6], dually chordal graphs [12], 2K 2 -free graphs [10] and more [4,21]. We refer the readers to a survey [18] on Gallai's question for more information.…”

Section: Introductionmentioning

confidence: 99%

Bonds intersecting long paths in $k$-connected graphs

Wei¹,

H²,

Zhao³

2022

Preprint

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A well-known question of Gallai (1966) [9] asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs [1,6,13,17], the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex v if and only if it meets the vertex-bond with respect to v. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let p denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least p − 1. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least p − 2. For a k-connected graph (k ≥ 3), we show that there is a bond meeting all paths of length at least p − t + 1, where t = k−2 2 if p is even and t = k−2 2 if p is odd. Our results provide analogs of the corresponding results of P. Wu [15] and S. McGuinness [22, Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.

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“…Identifying parts of the graph that are responsible for an increase in a graph parameter gives useful information about the graph structure and has been the central question around many graph problems. For instance, if the parameter in question is the size of a longest path, d = 1 and the operation is vertex deletion, the problem becomes equivalent to testing whether there exists a set of k vertices that intersects every longest path of the input graph [7][8][9]33]. Another example is that of computing the Hadwiger number of a graph.…”

Section: Introductionmentioning

confidence: 99%

Reducing graph transversals via edge contractions

Lima

Souza

2021

Journal of Computer and System Sciences

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For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k, d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection H according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in H, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Graph algorithms analysis; Theory of computation → Parameterized complexity and exact algorithms Keywords and phrases blocker problem, edge contraction, graph transversal, parameterized complexity, vertex cover, feedback vertex set, odd cycle transversal

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Intersection of longest paths in graph classes

Cited by 19 publications

References 15 publications

Transversals of Longest Cycles in Partial $k$-Trees and Chordal Graphs

Transversals of Longest Cycles in Partial $k$-Trees and Chordal Graphs

Bonds intersecting long paths in $k$-connected graphs

Reducing graph transversals via edge contractions

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