Let la(G) be the invariant introduced by Colin de Verdiere [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 Kn×T, where Kn 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#R##N##R##N#Part of this work was done while the author was visiting Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544.
['Hein van der Holst']
On the “largeur d'arborescence”