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 ⊆ 2^N, and real edge-weights w(e) for e ∈ E. Given a convex n-gon P in the plane with vertices x ₀, x ₁, ..., x _n − 1 which are arranged in this order clockwisely, we let each node i ∈ N correspond to the vertex x _i and define the area A _P (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 c _H (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.

• A _P (H) ≤ A _P (H′) for all convex n-gons P.

• c _H (i,j,k) ≤ c _H′(i,j,k) for all convex three-cuts C(i,j,k).