Abstract
A graph with forbidden transitions is a pair (G,F G ) where G: = (V G ,E G ) is a graph and F G is a subset of the set \( \{ (\{y,x\},\{x,z\}) \in E_G^2 \}.\) A path in a graph with forbidden transitions (G,F G ) is a path in G such that each pair ({y,x},{x,z}) of consecutive edges does not belong to F G . It is shown in [S. Szeider, Finding paths in graphs avoiding forbidden transitions, DAM 126] that the problem of deciding the existence of a path between two vertices in a graph with forbidden transitions is Np-complete. We give an exact exponential time algorithm that decides in time O(2n·n 5·log(n)) whether there exists a path between two vertices of a given n-vertex graph with forbidden transitions. We also investigate a natural extension of the minimum cut problem: we give a polynomial time algorithm that computes a set of forbidden transitions of minimum size that disconnects two given vertices (while in a minimum cut problem we are seeking for a minimum number of edges that disconnect the two vertices). The polynomial time algorithm for that second problem is obtained via a reduction to a standard minimum cut problem in an associated allowed line graph.
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Kanté, M.M., Laforest, C., Momège, B. (2013). An Exact Algorithm to Check the Existence of (Elementary) Paths and a Generalisation of the Cut Problem in Graphs with Forbidden Transitions. In: van Emde Boas, P., Groen, F.C.A., Italiano, G.F., Nawrocki, J., Sack, H. (eds) SOFSEM 2013: Theory and Practice of Computer Science. SOFSEM 2013. Lecture Notes in Computer Science, vol 7741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35843-2_23
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DOI: https://doi.org/10.1007/978-3-642-35843-2_23
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