Non-empty intersection of longest paths in H-free graphs
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We 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. We also show that if the order of a connected graph G is large relative to its connectivity κ(G), and its independence number α(G) satisfies α(G) ≤κ(G) + 2, then each vertex of maximum degree forms a longest path transversal of size 1.
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