w-density and w-balanced property of weighted graphs

The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced. A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it. In this paper,a good characterization of w-balanced weighted graphs is given. Applying this characterization,many large w-balanced weighted graphs are formed by combining smaller ones. In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed. It is shown that the w-density theory is closely related to the study of SEW(G,w) games.