CFHealthHub: Complex intervention to support adherence to treatment in adults with cystic fibrosis: external pilot trial

E-Poster Session

Distance and the pattern of intra-European trade

Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the element-connectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) element-disjoint Steiner forests, where h = | i Ti|. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests. • We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [12] in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future

Graphs without large triangle free subgraphs

AbstractThe main aim of the paper is to show that for 2⩽r<s and large enough n, there are graphs of order n and clique number less than s in which every set of vertices, which is not too small, spans a clique of order r. Our results extend those of Erdo&#x030B;s and Rogers

Non-conformable subgraphs of non-conformable graphs

AbstractWe show that if G and H are non-conformable graphs, with H being a subgraph of G of the same maximum degree Δ(G), and if Δ(G)⩾⌈12|V(G)|⌉, then |V(H)|=|V(G)|. We also show that this inequality is best possible, for when Δ(G)=⌊12|V(G)|⌋ there are examples of graphs G and H with Δ(H)=Δ(G) and |V(H)|<|V(G)| which are both non-conformable. We determine all such examples. Interest in this stems from the modified Conformability Conjecture of Chetwynd, Hilton and Hind, which would characterize all graphs G with Δ(G)⩾⌈12|V(G)|⌉, for which the total chromatic number χT(G) satisfies χT(G)=Δ(G)+1, in terms of non-conformable subgraphs