A new separation theorem with geometric applications

Let G = (V (G), E(G)) be an undirected graph with a measure function μ assigning non-negative values to subgraphs H so that μ(H) does not exceed the clique cover number of H. When μ satisfies some additional natural conditions, we study the problem of separating G into two subgraphs, each with a measure of at most 2μ(G)/3 by removing a set of vertices that can be covered with a small number of cliques G. When E(G) = E(G1) ∩ E(G2), where G1 = (V (G1), E(G1)) is a graph with V (G1) = V (G), and G2 = (V (G2), E(G2)) is a chordal graph with V (G2) = V (G), we prove that there is a separator S that can be covered with O( � lμ(G)) cliques in G, where l = l(G,G1) is a parameter similar to the bandwidth, which arises from the linear orderings of cliques covers in G1. The results and the methods are then used to obtain exact and approximate algorithms which significantly improve some of the past results for several well known NP-hard geometric problems. In addition, the methods involve introducing new concepts and hence may be of an indepandant interest.