Output-Sensitive Reporting of Disjoint Paths

Abstract.

A k -path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper we study the problem of performing k -path queries, with $k \leq 3$ , in a graph G with n vertices. We denote with $\ell$ the total length of the reported paths. For $k \leq 3$ , we present an optimal data structure for G that uses O(n) space and executes k -path queries in output-sensitive $O(\ell)$ time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st ) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.