A non-empty DS must possess a perfect matching (Figure ). A matching is a subgraph in which no two edges share a common node. Equivalently, a matching is a subgraph in which all nodes have degree one. A perfect matching Figure : Perfect Matching. A matching is a subgraph in which all nodes have degree one. A perfect matching contains all of the nodes of its parent. Nodes and edges of a matching are noted with black circles and hashed lines, respectively. The five-membered cycle on the left has no perfect matching. Such graphs are not valid delocalization subgraphs. The six-membered cycle on the right has a perfect matching, and would be a valid delocalization subgraph. Either graph or tree representation can be used to construct a perfect matching, but published algorithms use graphs.