Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that e...

We present a fixed parameter algorithm that constructively solves the k-dominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ n). In fact, we present our algorithm for any H-minor-free graph where H is a single-crossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)-minor-free graphs as a special case. As a...

Grigni and Hung [10] conjectured that H-minor-free graphs have (1 + )-spanners that are light, that is, of weight g(|H|, ) times the weight of the minimum spanning tree for some function g. This conjecture implies the efficient polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in H-minor free graphs; that is, a PTAS whose running time is of the form 2f( )nO(1) for...

Let A(G) be the adjacency matrix of a graph G. The largest eigenvalue of A(G) is called spectral radius of G. In this paper, an upper bound of spectral radii of K2,3-minor free graphs with order n is shown to be 3 2 + √ n− 7 4 . In order to prove this upper bound, a structural characterization of K2,3-minor free graphs is presented in this paper.

Lih, Wang and Zhu [Discrete Math. 269 (2003), 303–309] proved that the chromatic number of the square of a K4-minor free graph with maximum degree ∆ is bounded by ⌊3∆/2⌋+1 if ∆ ≥ 4, and is at most ∆+3 for ∆ ∈ {2, 3}. We show that the same bounds hold for the list chromatic number of squares of K4-minor free graphs. The same result was also proved independently by Hetherington and Woodall.

We show that problems that have finite integer index and satisfy a requirement we call treewidth-bounding admit linear kernels on the class of H-topological-minor free graphs, for an arbitrary fixed graph H . This builds on earlier results by Bodlaender et al. on graphs of bounded genus [2] and by Fomin et al. on H-minor-free graphs [9]. Our framework encompasses several problems, the prominent...

The cell surface complex (Detering et al., 1977, J. Cell Biol. 75, 899-914) of the sea urchin egg consists of two subcellular organelles: the plasma membrane, containing associated peripheral proteins and the vitelline layer, and the cortical vesicles. We have now developed a method of isolating the plasma membrane from this complex and have undertaken its biochemical characterization. Enzymati...

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs the tree t-spanner problem (the problem to decide whether G admits a tree t-spanner) is polynomial time solvable for ...

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is ...