Ear decomposition with bounds on ear length

Abstract

An ear decomposition for a graph G is a sequence of paths ( P 1, P 2,…, P k ) such that if, for each i, G i = G− P 1−⋯− P i , then P i is a maximal path of degree-two vertices in G i−1 whose endpoint(s) in G i have degree at least two, and G k is a union of disjoint cycles. Ear decomposition has become a standard tool in the design of graph algorithms, especially parallel connectivity algorithms. We show that the existence of two natural variations on ear decomposition can be tested in polynomial time. The first is a weak long-ear decomposition, in which every ear (path) is either disconnecting or is at least as long as a given bound B. The second is a strong short-ear decomposition, in which every ear is non-disconnecting and has length at most B.