Representation of short distances in structurally sparse graphs

04/19/2022

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by   Zdeněk Dvořák, et al.

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A partial orientation H⃗ of a graph G is a weak r-guidance system if for any two vertices at distance at most r in G, there exists a shortest path P between them such that H⃗ directs all but one edge in P towards this edge. In case H⃗ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most r in G. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.