Comparing hypergraphs by areas of hyperedges drawn on a convex polygon

Abstract

Let H = (N, E, w) be a hypergraph with a node set N = {0,1,...,n-1}, a hyperedge set E ⊆ 2N, and real edge-weights w(e) for e ∈ E. Given a convex n-gon P in the plane with vertices x0, x 1,...,xn-1 which are arranged in this order clockwisely, we let each node i ∈ N correspond to the vertex xi and define the area Ap(H) of H on P by the sum of weighted areas of convex hulls for all hyperedges in H. For 0 ≤ i < j < k ≤ n-1, a convex three-cut C(i, j, k) of N is {{i,..., j-1}, {j,...,k-1}, {k,...,n - 1,0,...,i - 1}} and its size cH(i,j,k) in H is defined as the sum of weights of edges e ∈ E such that e contains at least one node from each of {i,..., j-1}, {j,...,k - 1} and {k,..., n - 1,0,...,i-1}. We show that for two hypergraphs H and H′ on N, the following two conditions are equivalent. - Ap(H) ≤ Ap(H′) for all convex n-gons P. - cH(i,j,k) ≤ cH,(i,j,k) for all convex three-cuts C(i,j,k). © Springer-Verlag Berlin Heidelberg 2003.