For a set T of rooted binary leaf-labelled trees, we present an algorithm that finds all of the minor-minimal trees that are compatible with T . The running time of this algorithm is polynomial up to the number of trees with this property. This type of problem arises in several areas of classification, particularly evolutionary biology.

Let a, b, c, d be four vertices in a graph G. A K4-minor rooted at a, b, c, d consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of G, respectively containing a, b, c, d. We characterise precisely when G contains a K4-minor rooted at a, b, c, d by describing six classes of obstructions, which are the edge-maximal graphs containing no K4-minor rooted at a, b, c, d. The foll...

Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence; for most of these a simple test can be used to quickly show that the form of the asymptotics is the same as that for the class of rooted trees: C · ρ−n · n−3/...

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (Gi) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of Gi is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the case where the sequence (Gi) is replace...

A biconnected plane graph G is called internally triconnected if any cut-pair consists of outer vertices and its removal results in only components each of which contains at least one outer vertex. In a rooted plane graph, an edge is designated as an outer edge with a specified direction. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) (resp., Gint(n, g)) denote the class of all trico...