Good characterizations for some degree constrained subgraphs

The degree constrained subgraph problem is to find a subgraph of a graph with degrees as close to a given collection of degree prescriptions as possible. This problem is NP-complete in general, but for the case when no prescription contains two consecutive gaps, Lovász gave a structural description, and Cornuéjols gave a polynomial algorithm. However, compact good characterizations are known only in some special cases, such as parity intervals or general antifactors. The main result of the present paper is a simple good characterization for the special case when for every prescription it holds that all gaps have the same parity. This class contains most cases where compact good characterizations were known. The technique we apply is replacing the vertices by certain subgraphs, called gadgets—a method developed by Tutte for showing how the simple b-matching problem can be reduced to classical matchings. For this class, using a result of Pap, this approach yields the polynomiality of the edge weighted degree constrained subgraph problem.