On the Realization of Degree Sequences in Parallel

When is a sequence of integers realizable as the degree sequence of a graph? There are well known necessary and suucient conditions for a sequence of integers to constitute degrees of a graph 1]. There exists a sequential linear time algorithm that construct a graph from a given degree sequence. This algorithm however seems to be inherenty sequencial, and thus hard to parallelize. In this paper we prove that the decision version of the degree sequence problem is in NC. 1 Introduction Let G be an undirected graph with an integer capacity f(v) associated with each node v. A subset M of the set of edges of G is called a f-matching if and only if each vertex v in G is incident to at most f(v) edges in M. In particular each (usual) matching is a 1-matching. The maximum cardinality f-matching problem is a very natural generalization of the maximum matching problem. In turn, we can reduce the maximum f-matching problem to the maximum matching problem 3] by following the reduction of the f-factor problem to the perfect matching problem due to Tutte 4]. Fact 1.1 Lingas 93] A maximum f-matching in a graph on n nodes and m edges can be constructed by randomized algorithms, sequentially in time O(n 2 m), and in parallel in time O(log 2 n) using a CRCW PRAM with O(n 8 m) processors 3]. In this paper we are going to look at a special case of the maximum cardinality f-matching. Deenition 1.1 Given a sequence of integers d 1 d 2 d n , construct a graph G whose vertices v i satisfy deg(v i) = d i. We call this problem the degree sequence problem (DSP for short). It is not diicult to see that the degrees sequence problem is the maximum f-matching problem restricted to complete graphs. Fact 1.1 implies that this problem is in RNC. It is well known 1] that a given sequence d 1 d 2 d n is the degree sequence of a graph if and only if: P n i=1 d i is even and P k i=1 d i k(k ? 1) + P n i=k+1 minfk; d i g holds for each 1 k n.