When bad things happen to good trees*

Abstract

When the edges in a tree or rooted tree fail with a certain ®xed probability, the (greedoid) rank may drop. We compute the expected rank as a polynomial in p and as a real number under the assumption of uniform distribution. We obtain several different expressions for this expected rank polynomial for both trees and rooted trees, one of which is especially simple in each case. We also prove two extremal theorems that determine both the largest and smallest values for the expected rank of a (rooted or unrooted) tree, and precisely when these extreme bounds are achieved. We conclude with directions for further study. ß

Distinguished vertices in probabilistic rooted graphs

“…More motivation and background can © 2009 Wiley Periodicals, Inc. be found in [4]. This work continues the study initiated in [1] and [4].…”

“…The fact that so much freedom is allowed for vertex location in these cases indicates how difficult is the general problem of vertex location. This is explored in some detail for other families of graphs in [1,4,10,11].…”

Pruning processes and a new characterization of convex geometries

“…Our work and in particular the implication 1 → 3 of our main Theorem 3.1 is related to previous work of Aivaliotis, Gordon, and Graveman [2] and Gordon [10]. For more information on this connection, see Section 3.1.…”

Non-isomorphic caterpillars with identical subtree data

“…This amounts to creating two trees in which t 1 Further, we could create additional counterexamples by modifying one of the factors in this expression. The reader can check that the generating polynomials in Table 1 also produce non-isomorphic caterpillars with the same greedoid Tutte polynomial.…”