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44
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
The strong perfect graph theorem
 ANNALS OF MATHEMATICS
, 2006
"... A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asse ..."
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Cited by 166 (15 self)
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A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornuéjols and Vuˇsković — that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge’s conjecture cannot have either of these properties). In this paper we prove both these conjectures.
Combinatorial Optimization: Packing and Covering
, 2000
"... The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optim ..."
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Cited by 40 (1 self)
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The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Minmax theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathematics. In addition to minmax and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition ” results. In these notes, we present several of these beautiful results. Three chapters cover minmax and polyhedral results. The next four cover excluded minor results. In the last three, we
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 36 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
... Minors In Graphs Of Bounded TreeWidth
 J. Combin. Theory Ser. B
, 2000
"... It is shown that for any positive integers k and w there exists a ..."
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Cited by 10 (3 self)
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It is shown that for any positive integers k and w there exists a
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
Generating Internally FourConnected Graphs
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. A graph G is internally 4connected if it is simple, 3connected, has at least five vertices, and if for every partition (A; B) of the edgeset of G, either jAj 3, or jBj 3, or at leas ..."
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Cited by 6 (1 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. A graph G is internally 4connected if it is simple, 3connected, has at least five vertices, and if for every partition (A; B) of the edgeset of G, either jAj 3, or jBj 3, or at least four vertices of G are incident with an edge in A and an edge in B. We prove that if H and G are internally 4connected graphs such that they are not isomorphic, H is a minor of G and they do not belong to a family of exceptional graphs, then there exists a graph H 0 such that H 0 is isomorphic to a minor of G and either H 0 is obtained from H by splitting a vertex, or H 0 is an internally 4connected graph obtained from H by means of one of four possible constructions. This is a first step toward a more comprehensive theorem along the same lines.
Excluding a Countable Clique
, 1998
"... We extend the excluded K n minor theorem of Robertson and Seymour to infinite graphs, and deduce a structural characterization of the infinite graphs that have no K #0 minor. The latter is a refinement of an earlier characterization of Robertson, Seymour and the second author. ..."
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Cited by 4 (0 self)
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We extend the excluded K n minor theorem of Robertson and Seymour to infinite graphs, and deduce a structural characterization of the infinite graphs that have no K #0 minor. The latter is a refinement of an earlier characterization of Robertson, Seymour and the second author.
Obstruction Sets For OuterProjectivePlanar Graphs
 Ars Combinatoria
"... . A graph G is outerprojectiveplanar if it can be embedded in the projective plane so that every vertex appears on the boundary of a single face. We exhibit obstruction sets for outerprojectiveplanar graphs with respect to the subdivision, minor, and Y \Delta orderings. Equivalently, we find th ..."
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Cited by 3 (2 self)
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. A graph G is outerprojectiveplanar if it can be embedded in the projective plane so that every vertex appears on the boundary of a single face. We exhibit obstruction sets for outerprojectiveplanar graphs with respect to the subdivision, minor, and Y \Delta orderings. Equivalently, we find the minimal nonouterprojectiveplanar graphs under these orderings. x1 Introduction The most frequently cited [B] result in graph theory is Kuratowski's Theorem [K], which states that a graph is planar if and only if it does not contain a subdivision of either K 5 or K 3;3 . This is an example of an obstruction theorem; a characterization of graphs with a particular property in terms of excluded subgraphs. Obstruction theorems may involve other properties besides planarity and other orderings besides the subgraph order. Let P be a property of graphs, formally, P is some collection of graphs. Let be a partial ordering on all graphs. We say that P is hereditary under if G 2 P and H G im...