Optimization and recognition for K5-minor free graphs in linear time

Abstract

We present a linear time algorithm which determines whether an input graph contains K 5 as a minor and outputs a K 5-model if the input graph contains one. If the input graph has no K 5-minor then the algorithm constructs a tree decomposition such that each node of the tree corresponds to a planar graph or a graph with eight vertices. Such a decomposition can be used to obtain algorithms to solve various optimization problems in linear time. For example, we present a linear time algorithm for finding an seperator and a linear time algorithm for solving k-realisation on graphs without a K 5-minor. Our algorithm will also be used, in a separate paper, as a key subroutine in a nearly linear time algorithm to test for the existence of an H-minor for any fixed H. © 2008 Springer-Verlag Berlin Heidelberg.