Listing Small Minimal Separators of a Graph

12/16/2020

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by   Tuukka Korhonen, et al.

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Let G be a graph and a,b vertices of G. A minimal a,b-separator of G is an inclusion-wise minimal vertex set of G that separates a and b. We consider the problem of enumerating the minimal a,b-separators of G that contain at most k vertices, given some integer k. We give an algorithm which enumerates such minimal separators, outputting the first R minimal separators in at most poly(n) R ·min(4^k, R) time for all R. Therefore, our algorithm can be classified as fixed-parameter-delay and incremental-polynomial time. To the best of our knowledge, no algorithms with non-trivial time complexity have been published for this problem before. We also discuss barriers for obtaining a polynomial-delay algorithm.