Random sampling of minimally cyclic digraphs with given imbalance sequence

The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the outdegree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic digraph as a connected digraph which is either acyclic or has exactly one oriented cycle whose removal disconnects the digraph. In this paper we consider the problem of sampling minimally cyclic digraphs with a given imbalance sequence. We present an algorithm for generating all minimally cyclic graphs realizing an imbalance sequence with each realization produced in linear time. We then construct a Markov chain on the set M of all minimally cyclic digraphs with a given imbalance sequence and prove that this Markov chain is ergodic, reversible and rapidly mixing. Thus we can sample through M in polynomial time. 2010 Mathematics Subject Classification: 05C20, 05C07, 60J10, 60J15, 65C05