Optimal grid representations

Abstract

AbstractA graph G is a grid intersection graph if G is the intersection graph of ℋ︁ ∪ ℐ, where ℋ︁ and ℐ are, respectively, finite families of horizontal and vertical linear segments in the plane such that no two parallel segments intersect. (This definition implies that every grid intersection graph is bipartite.) The family ℋ︁ ∪ ℐ is a representation of G. As a consequence of a characterization of grid intersection graphs by Kratochvíl, we observe that when a bipartite graph G = (U ∪ W, E) with minimum degree at least two is a grid intersection graph, then there exists a normalized representation of G on the (r × s)‐grid for r = |U| and s = |W|, that is, a representation in which all end points of segments have integer‐valued coordinates belonging to {(x, y) ∈ N × N | 1 ≤ y ≤ r, 1 ≤ x ≤ s} and the representative segment of each vertex lies on a distinct horizontal or vertical line. A natural problem, with potential applications to circuit layout, is the following: among all the possible normalized representations of G, find a representation ℛ such that the sum of the lengths of the segments in ℛ is minimum. In this work we introduce this problem and present a mixed integer programming formulation to solve it. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 187–193 2004