Transversals of Longest Paths and Cycles

Dieter Rautenbach (1) , Jean-Sébastien Sereni (2)

1 Institut für Optimierung und Operations Research
2 ORPAILLEUR - Knowledge representation, reasonning

Let G be a graph of order n. Let lpt(G) be the minimum cardinality of a set X of vertices of G such that X intersects every longest path of G and define lct(G) analogously for cycles instead of paths. We prove that (i) lpt(G) ≤ ceiling(n/4-n^{2/3}/90), if G is connected, (ii) lct(G) ≤ ceiling(n/3-n^{2/3}/36), if G is 2-connected, and (iii) lpt(G) ≤ 3, if G is a connected circular arc graph. Our bound on lct(G) improves an earlier result of Thomassen and our bound for circular arc graphs relates to an earlier statement of Balister et al. the argument of which contains a gap. Furthermore, we prove upper bounds on lpt(G) for planar graphs and graphs of bounded tree-width.

Mots clés: Longest path; longest cycle; transversal

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Dieter Rautenbach, Jean-Sébastien Sereni. Transversals of Longest Paths and Cycles. [Research Report] Loria & Inria Grand Est. 2013. ⟨hal-00793271⟩