Optimal grid representations

• Autor(a) principal: Fampa, M. H. C.
• Data de Publicação: 2004
• Outros Autores: Klein, S., Protti, F., Rêgo, D. C. A.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFRJ
• Texto Completo: http://hdl.handle.net/11422/2582
• topic: CNPQ::ENGENHARIAS::ENGENHARIA ELETRICA::TELECOMUNICACOES; Intersection graph of segments; Grid intersection graph; Grid representation; Integer programming

A graph G is a grid intersection graph if G is the intersection graph of ℋ ∪ ℐ, where ℋ 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 ℋ ∪ 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.