On the “largeur d'arborescence”

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