 School of Mathematics and Statistics  Research Publications
School of Mathematics and Statistics  Research Publications
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ItemA LinearTime Algorithm to Find a Separator in a Graph Excluding a MinorReed, B ; Wood, DR (ASSOC COMPUTING MACHINERY, 20091001)Let G be an n vertex m edge graph with weighted vertices. A pair of vertex sets A , B ⊆ V ( G ) is a 2/3 separation of order  A ∩ B  if A ∪ B = V ( G ), there is no edge between A − B and B − A , and both A − B and B − A have weight at most 2/3 the total weight of G . Let ℓ ∈ Z + be fixed. Alon et al. [1990] presented an algorithm that in O ( n 1/2 m ) time, outputs either a K ℓ minor of G , or a separation of G of order O ( n 1/2 ). Whether there is a O ( n + m )time algorithm for this theorem was left as an open problem. In this article, we obtain a O ( n + m )time algorithm at the expense of a O ( n 2/3 ) separator. Moreover, our algorithm exhibits a tradeoff between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a K ℓ minor of G , or a separation of G with order O ( n (2−ϵ)/3 in O ( n 1 + ϵ + m ) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no K ℓminor.