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A graph is a minor of another if the first can be obtained from a sub-graph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and… (More)
Given a 0-1 square matrix A, when can some of the 1's be changed to −1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When… (More)
I give a self-contained exposition of a linear-time planarity algorithm of Shih and Hsu.
We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices ;', j such that i < j and G¡ is isomorphic to a minor of Gj. This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G\ cannot be omitted by our earlier result. The proof is complex and… (More)
For each infinite cardinal k we give several necessary and sufficient conditions for a graph not to contain a minor isomorphic to the infinite k-branching tree in terms of a certain kind of a "tree-decomposition," in terms of a "path-decomposition," and also in terms of a "cops-and-robber game." We also give necessary and sufficient conditions for a graph… (More)
A hypergraph H has tree-width k (a notion introduced by Robertson and Seymour) if k is the least integer such that H admits a tree-decomposition of tree-width k. We prove a compactness theorem for this notion, that is, if every finite subhypergraph of H has tree-width < k, then H itself has tree-width < k. This result will be used in a later paper on… (More)
Granulopoiesis-related genes are distinctively upregulated in peripheral leukocytes of patients with antineutrophil cytoplasmic autoantibodies (ANCA)-associated glomerulonephritis. Affymetrix microarrays identified the upregulation of nine neutrophilic primary granule genes, including myeloperoxidase (MPO) and proteinase 3 (PR3), plus five secondary granule… (More)
We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, link-less embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture , and Pfaaan orientations of bipartite graphs.