Induced Disjoint Paths in Circular-Arc Graphs in Linear Time

Abstract

AbstractThe Induced Disjoint Paths problem is to test whether a graph \(G\) with \(k\) distinct pairs of vertices \((s_{i},t_{i})\) contains paths \(P_{1},\ldots ,P_{k}\) such that \(P_{i}\) connects \(s_{i}\) and \(t_{i}\) for \(i=1,\ldots ,k\), and \(P_{i}\) and \(P_{j}\) have neither common vertices nor adjacent vertices (except perhaps their ends) for \(1 \le i < j \le k\). We present a linear-time algorithm that solves Induced Disjoint Paths and finds the corresponding paths (if they exist) on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs \((s_{i},t_{i})\) are not necessarily distinct.KeywordsDisjoint Paths ProblemInterval GraphsLinear Time AlgorithmTerminal PairTerminal PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.