Transversals of longest paths

Abstract

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.

$K_2$-Hamiltonian Graphs: I

“…non-traceable, but every vertex-deleted subgraph is traceable) also provides a negative answer to Gallai's question. On the other hand, substantial efforts have been made to prove that in certain classes of graphs we can guarantee that the intersection of all longest paths is non-empty, see [6] for a recent contribution.…”

Transversals of longest cycles in partial k‐trees and chordal graphs

“…Transversals of longest paths have been also studied [4,21]. Also, other questions about intersection of longest cycles have been raised by several authors [5,17,19,23].…”

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

“…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].…”