Efficient enumeration of all minimal separators in a graph

Abstract

This paper presents an efficient algorithm for enumerating all minimal a- b separators separating given non-adjacent vertices a and b in an undirected connected simple graph G = ( V, E), Our algorithm requires O( n 3 R ab ) time, which improves the known result of O( n 4 R ab ) time for solving this problem, where ¦V¦= n and R ab is the number of minimal a- b separators. The algorithm can be generalized for enumerating all minimal A- B separators that separate non-adjacent vertex sets A, B < V, and it requires O( n 2( n − n A − n b ) R AB ) time in this case, where n a = ¦A¦, n B = ¦B¦ and r AB is the number of all minimal A− B separators. Using the algorithm above as a routine, an efficient algorithm for enumerating all minimal separators of G separating G into at least two connected components is constructed. The algorithm runs in time O( n 3 R + Σ + n 4 R Σ ), which improves the known result of O( n 6 R Σ ) time, where R σ is the number of all minimal separators of G and R Σ </ R + Σ = ∑ 1</i≠j</n, (v i, v j) ∉ E R v iv j </ (n (n − 1) 2 − m)R Σ . Efficient parallelization of these algorithms is also discussed. It is shown that the first algorithm requires at most O(( n log n )R ab) time and the second one runs in time O(( n log n )R + Σ+n log nR Σ) on a CREW PRAM with O( n 3) processors.