On the “largeur d'arborescence”

✍ Scribed by Hein van der Holst

• Book ID: 102341548
• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 271 KB
• Volume: 41
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.10046

✦ Synopsis

Abstract

Let la(G) be the invariant introduced by Colin de Verdière [J. Comb. Theory, Ser. B., 74:121–146, 1998], which is defined as the smallest integer n≥0 such that G is isomorphic to a minor of K~n~×T, where K~n~ is a complete graph on n vertices and where T is an arbitrary tree. In this paper, we give an alternative definition of la(G), which is more in terms of the tree‐width of a graph. We give the collection of minimal forbidden minors for the class of graphs G with la(G)≤k, for k=2, 3. We show how this work on la(G) can be used to get a forbidden minor characterization of the graphs with $\nu^{\cal R}_1$(G)≤3. Here, $\nu^{\cal R}_1$(G) is another graph parameter introduced in the above cited paper. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 24–52, 2002