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Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph 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 disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph 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 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Progress on Perfect Graphs
, 2003
"... A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restr ..."
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Cited by 7 (3 self)
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A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first
Recent Excluded Minor Theorems
 Surveys in Combinatorics, LMS Lecture Note Series
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
EXCLUDING MINORS IN NONPLANAR GRAPHS OF GIRTH AT LEAST FIVE
, 1999
"... A graph is quasi 4connected if it is simple, 3connected, has at least five vertices, and for every partition (A, B, C) of V(G) either C≥4, or G has an edge with one end in A and the other end in B, orone of A,B has at most one vertex. We show that any quasi 4connected nonplanar graph with minim ..."
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Cited by 3 (1 self)
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A graph is quasi 4connected if it is simple, 3connected, has at least five vertices, and for every partition (A, B, C) of V(G) either C≥4, or G has an edge with one end in A and the other end in B, orone of A,B has at most one vertex. We show that any quasi 4connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P − 10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte’s Four Flow Conjecture: every 2edge connected graph with no minor isomorphic to P − 10 has a nowherezero 4flow. This extends a result of Kilakos and Shepherd who proved the same for 3regular graphs.