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

Let lct ( G ) be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2‐connected graph G. We show that lct ( G ) ≤ k − 1 if G is a partial k‐tree and that lct ( G ) ≤ max { 1 , ω ( G ) − 3 } if G is chordal, where ω ( G ) is the cardinality of a maximum clique in G. Those results imply that all longest cycles intersect in 2‐connected series‐parallel graphs and in 3‐trees.