The maximum Wiener index of maximal planar graphs

The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

Uma proposta de solução aproximada para o problema do subgrafo planar de peso máximo

A proposta deste trabalho consiste em uma solução aproximada para o problema do subgrafo planar de peso máximo (WMPG Weighted Maximal Planar Graph). O algoritmo baseia-se na adição de vértices, aproveitando-se da construção de triangulações nas faces do grafo. A vantagem do uso deste algoritmo dá-se pelo fato que todo grafo gerado por ele é maximal planar, descartando a necessidade de um teste de planaridade. Apresentamos um algoritmo sequencial e um paralelo para o problema WMPG e suas respectivas implementações. Os resultados obtidos com a versão paralela executando em uma arquitetura manycore, com instâncias de até 85 vértices, apresentaram speedups de até 107 vezes em relação à solução sequencial.

Facility Layout Planning

The weighted maximal planar graph (WMPG) appears in many applications. It is currently used to design facilities layout in manufacturing plants. Given an edge-weighted complete simple graph G, the WMPG involves finding a sub-graph of G that is planar in the sense that it could be embedded on the plane such that none of its edges intersect, and is maximal in the sense that no more edges can be added to it unless its planarity is violated. Finally, it is optimal in the sense that the resulting maximal planar graph holds the maximum sum of edge weights. In this chapter, the aim is to explain the application of planarity in facility layout design. The mathematical models and the algorithms developed for the problem so far are explained. In the meanwhile, the corollaries and theorems needed to explain the algorithms and models are briefly given. In the last part, an explanation on how to draw block layout from the adjacency graph is given.