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 paths reported. 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.