Connectivities for k-knitted graphs and for minimal counterexamples to Hadwiger\u27s Conjecture

For a given subset S subset of V (G) of a graph G, the pair (G, S) is knitted if for every partition of S into non-empty subsets S-1, S-2, ... , S-t, there are disjoint connected subgraphs C-1, C-2, ... , C-t in G so that S-i subset of C-i. A graph G is l-knitted if (G, S) is knitted for all S subset of V(G) with vertical bar S vertical bar = l. In this paper, we prove that every 9l-connected graph is l-knitted. Hadwiger\u27s Conjecture states that every k-chromatic graph contains a K-k-minor. We use the above result to prove that the connectivity of minimal counterexamples to Hadwiger\u27s Conjecture is at least k/9, which was proved to be at least 2k/27 in Kawarabayashi (2007) [4]. (C) 2013 Elsevier Inc. All rights reserved

Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs

We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topological subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input formula and the width of the poset. Our result for successor-invariant FO extends previous results for this logic on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors (Eickmeyer et al., LICS 2013), further narrowing the gap between what is known for FO and what is known for successor-invariant FO. The proof uses Grohe and Marx's structure theorem for graphs with excluded topological subgraphs. For order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO carries over to order-invariant FO