Abstract
The 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.
This work is supported by EPSRC (EP/K025090/1) and Royal Society (JP100692). The research leading to these results has also received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 267959.
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Notes
- 1.
Due the space restrictions some proofs in this section and in the next ones are omitted or sketched. The full paper, with complete proofs, can be found in [10].
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Golovach, P.A., Paulusma, D., van Leeuwen, E.J. (2014). Induced Disjoint Paths in Circular-Arc Graphs in Linear Time. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_19
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