The complexity of two graph orientation problems

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol

Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects