Fractional support graph of a matching relaxation ($H=(V,\{e:0<x_e<1\})$)

For the degree-constrained linear programming relaxation of a matching in a graph, retain precisely the edges with fractional coordinates. In a bipartite graph, an even graph cycle in this subgraph permits alternating positive and negative perturbations preserving all vertex sums. If the subgraph is a forest, use a graph path between two tree leaves: its endpoints have positive slack and its internal sums remain unchanged. Either perturbation expresses the original feasible point as the midpoint of two distinct feasible points. Hence a fractional feasible point cannot be an extreme point, proving integrality of all extreme points of the bipartite degree relaxation.