Optimization and Recognition for K 5-minor Free Graphs in Linear Time

Abstract

We present a linear time algorithm which determines whether an input graph contains K5 as aminor and outputs a K5-model if the input graph contains one. If the input graph has no K5-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 O(√n) seperator and a linear time algorithm for solving k-realisation on graphs without a K5-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.