Transversals of Longest Paths and Cycles

TL;DR

This paper establishes new upper bounds on the minimum vertex set intersecting all longest paths and cycles in various classes of graphs, improving previous results and addressing gaps in earlier proofs.

Contribution

It provides improved bounds for lpt(G) and lct(G) in connected, 2-connected, circular arc, planar, and bounded tree-width graphs, advancing understanding of graph transversals.

Findings

lpt(G)  (n/4 - n^{2/3}/90) for connected graphs

lct(G)  (n/3 - n^{2/3}/36) for 2-connected graphs

lpt(G)  3 for connected circular arc graphs

Abstract

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 lpt(G) \leq ceiling(n/4-n^{2/3}/90), if G is connected, lct(G) \leq ceiling(n/3-n^{2/3}/36), if G is 2-connected, and \lpt(G) \leq 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 \emph{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.