We consider a dynamic problem introduced by B. Hartnell in 1995 [4]. Let (G, r) be a connected rooted graph (where r ∈ V (G)). At time 0, a fire breaks out at r. At each subsequent time interval, the firefighter defends some vertex which is not on fire, and then the fire spreads to all undefended neighbours of each burning (i.e., on fire) vertex. Once defended, a vertex remains so for all time ...

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...