Nonempty Intersection of Longest Paths in $2K_2$-Free Graphs

Golan, Gili; Shan, Songling

doi:10.37236/7487

Cited by 13 publications

(16 citation statements)

References 18 publications

(20 reference statements)

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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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“…Chen [5] proved the same for graphs with matching number smaller than three, while Cerioli and Lima [4,17] proved it for P 4 -sparse graphs, (P 5 , K 1,3 )-free graphs, graphs that are the join of two other graphs and starlike graphs, a superclass of split graphs. Finally, Jobson et al [14] proved it for dually chordal graphs and Golan and Shan [11] for 2K 2 -free graphs. A more general approach to Gallai's question is to ask for the size of the smallest transversal of longest paths of a graph, that is, the smallest set of vertices that intersects every longest path.…”

Section: Introductionmentioning

confidence: 97%

Transversals of longest paths

Cerioli

Fernandes

Gómez

et al. 2020

Discrete Mathematics

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Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Proposition 4. Let t be a node of T and v be a vertex of G such thatthen t ′ and t ′′ are in the same branch of T at t. Proposition 5. Let u and v be two vertices of G, and let t be a node of T. If u, v /∈ V t , and u and v are not separated by V t in G, then Branch t (u) = Branch t (v). Proposition 6. Let t be a node of T and P be a path fenced by V t . For every two vertices u and v in P − V t ,Proof. By definition of fenced paths, u and v lie in the same component of G − V t , so they are not separated by V t in G and we can apply Proposition 5.Proposition 7. If P is a path fenced by V t for some t ∈ V(T), then there exists a neighbor t ′ of t in T such that Branch t (P) = Branch t (t ′ ).Proof. Let u be a vertex of P − V t (that exists by the definition of fenced). As u /Proposition 8 appears in the book of Diestel [8] as Lemma 12.3.1. Proposition 9 is a corollary of Proposition 8. Proposition 8. Let pq ∈ E(T). Let T p = Proposition 9. Let pq ∈ E(T). Let u and v be vertices of G with, then u and v are not adjacent. Proposition 11. Let t ∈ V(T) and tt ′ ∈ E(T). Let P ′ be a path fenced by V t that 1-touches V t such thatProof. Suppose by contradiction that P ′ ∩ Q ′ = ∅. As P ∩ Q ∩ V t = ∅, we must have that Branch t (Q ′ ) = Branch t (P ′ ) = Branch t (t ′ ). By Proposition 10, V t ∩ Q ′ ⊆ V t ′ , a contradiction.

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“…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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Nonempty Intersection of Longest Paths in $2K_2$-Free Graphs

Cited by 13 publications

References 18 publications

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

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

Transversals of longest paths

Bonds intersecting long paths in $k$-connected graphs

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