Another Look at the Degree Constrained Subgraph Problem

There are several versions of the degree constrained subgraph problem, and we refer to the following: Given an undirected graph G = (V, E) with n vertices, and 2n integers al, . . . . a,,, br , . . . . b,, find a subgraph G’ = (V, E’) of G such that ai < dG’(vi) < bi for 1 < i < n and i E’l is maximized. Here d&vi) denotes the degree of vi restricted to G’. This problem has been solved by Urquhart [S], and a polynomial solution to it can also be derived from Edmonds and Johnson’s work [2]. Both papers use the linear programming approach. A more combinatorial approach is presented here. In Section 2 we solve a restricted problem in which ai = 0 for all i. This problem is reduced to the regular maximum matching problem via a simple construction IL. The same construction also yields a reduction of the weighted version of this problem to the weighted maximum matching problem. (In the weighted problem a weight w(e) is assigned to each e E E and ZeE E’ w(e) is maximized rather than 1 E' I.) In Section 3 an alternating path technique is used to obtain a solution to the general problem from that of the restricted problem. The corresponding weighted problem is reduced to the weighted matching problem 2.