Separators in Graphs with Negative and Multiple Vertex Weights
Djidjev, Hristo N.
A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show that any separator theorem implies various weighted separator theorems. In particular, we show that if the vertices of the graph have real-valued weights, which maybe positive or negative, then the graph can be divided exactly in half according to weight. If k unrelated sets of weights are given, the graph can be divided simultaneously by all sets of weights. These results considerably strengthen earlier results of Gilbert, Lipton, and Tarjan: (1) for k=1 with the weights restricted to be nonnegative, and (2) for k > 1, nonnegative weights, and simultaneous division within a factor of (1 + e) of exactly in half.