Sublinear Longest Path Transversals and Gallai Families
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We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit constant-size longest path transversals. The same technique allows us to show that 2-connected graphs admit sublinear longest cycle transversals. We also make progress toward a characterization of the graphs H such that every connected H-free graph has a longest path transversal of size 1. In particular, we show that the graphs H on at most 4 vertices satisfying this property are exactly the linear forests. Finally, we show that if the order of a connected graph G is large relative to its connectivity κ(G) and α(G) ≤κ(G) + 2, then each vertex of maximum degree forms a longest path transversal of size 1.
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