Optimization and Recognition for K5-minor Free Graphs in Linear Time

• Published in: LATIN 2008: Theoretical Informatics pp. 206 - 215
• Main Authors: Reed, Bruce, Li, Zhentao
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• ISBN: 9783540787723; 3540787720
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-540-78773-0_18

We present a linear time algorithm which determines whether an input graph contains K5 as a minor 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\sqrt{n})$\end{document} 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.