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 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(2 n ·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. © 2013 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
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 Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7741 LNCS, pp. 257–267). https://doi.org/10.1007/978-3-642-35843-2_23
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