The topological containment problem is known to be polynomial-time solvable for any fixed pattern graph H , but good characterisations have been found for only a handful of non-trivial pattern graphs. The complete graph on five vertices, K5, is one pattern graph for which a characterisation has not been found. The discovery of such a characterisation would be of particular interest, due to the ...

We prove that if N is an internally 4-connected minor of an internally 4-connected binary matroid M with E(N) ≥ 4, then there exist matroids M0,M1, . . . ,Mn such that M0 ∼= N , Mn = M , and, for each i ∈ {1, . . . , i}, Mi−1 is a minor of Mi, |E(Mi−1)| ≥ |E(Mi)| − 2, and Mi is 4-connected up to separators of size 5.

The synaptic connectivity of cortex is plastic, with experience shaping the ongoing interactions between neurons. Theoretical studies of spike timing-dependent plasticity (STDP) have focused on either just pairs of neurons or large-scale simulations. A simple analytic account for how fast spike time correlations affect both microscopic and macroscopic network structure is lacking. We develop a ...

We show that for every connected graph G of diameter ≥ 3, the graph G3 has average degree ≥ 7 4δ(G). We also provide an example showing that this bound is best possible. This resolves a question of Hegarty [3].

In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have a...

Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), ..., d(v, Sk)), where d(v, Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a re...

Lcclerc proved that the dimension of the partiailly ordered slet consisting of all subf~ce'~ of a tree T, m&red by inclusion, is the number of end yuints of 'I'. Leclerc posed the probkrn of determitAng the dimension the partially ed set P consisting of all inducxxI connected subgraphs of a connec graph G for tPisa lattice. In this paper, WC prove that the poset P consisting af all induced conn...

A graph is 1-planar if it can be drawn in the plane such that every edge crosses at most one other edge. A connected graph H is strongly light in a family of graphs G, if there exists a constant λ, such that every graph G in G contains a subgraph K isomorphic to H with degG(v) ≤ λ for all v ∈ V(K). In this paper, we present some strongly light subgraphs in the family of 1-planar graphs with min...