Non-Crossing Shortest Paths are Covered with Exactly Four Forests

10/24/2022

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by   Lorenzo Balzotti, et al.

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Given a set of paths P we define the Path Covering with Forest Number of P (PCFN(P)) as the minimum size of a set F of forests satisfying that every path in P is contained in at least one forest in F. We show that PCFN(P) is treatable when P is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if P is a set of non-crossing shortest paths of a planar graph G whose extremal vertices lie on the same face of G, then PCFN(P)≤4, and this bound is tight.