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The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... 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. Freem ..."
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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
A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
, 2008
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Graph and map isomorphism and all polyhedral embeddings in linear time
 IN PROCEEDINGS OF THE 40TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. Th ..."
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k≥3, we want to find an embedding of G in S of face width at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.
Upward Planar Drawing of Single Source Acyclic Digraphs
, 1990
"... A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We ..."
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A upward plane drawing of a directed acyclic graph is a straight line drawing in the Euclidean plane such that all directed arcs point upwards. Thomassen [30] has given a nonalgorithmic, graphtheoretic characterization of those directed graphs with a single source that admit an upward drawing. We present an efficient algorithm to test whether a given singlesource acyclic digraph has a plane upward drawing and, if so, to find a representation of one such drawing. The algorithm decomposes the graph into biconnected and triconnected components, and defines conditions for merging the components into an upward drawing of the original graph. For the triconnected components we provide a linear algorithm to test whether a given plane representation admits an upward drawing with the same faces and outer face, which also gives a simpler (and algorithmic) proof of Thomassen's result. The entire testing algorithm (for general single source directed acyclic graphs) operates in O(n²) time and...
Practical Toroidality Testing
 Proc. of the Eighth Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the gr ..."
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We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the graphs on up to 10 vertices and the 11 vertex ones up to 24 edges, and of these 3884 are topological obstructions, and 2249 are also minor order obstructions. A cursory search of 12 and 13 vertex graphs resulted in several more. We purport that this approach has proved practical as it has permitted us to compile what we believe to be the biggest collection of torus obstructions in the world to date. 1 Introduction A graph is said to be embedded on a surface if it is drawn there with no crossing edges. A graph is planar if it can be drawn on the sphere, and is toroidal if it can be drawn on the torus (a sphere with one handle). The genus of a planar graph is zero, and a nonplanar graph which ...
Graph Minors and Graphs on Surfaces
, 2001
"... Graph minors and the theory of graphs embedded in surfaces are ..."
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Graph minors and the theory of graphs embedded in surfaces are
Universal obstructions for embedding extension problems
"... Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded bran ..."
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Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded branch size up to a bounded number of “almost disjoint ” millipedes. Moreover, B contains a nice subgraph ˜ B with the following properties. If K is 2cell embedded in some surface and F is a face of K, then ˜ B admits exactly the same types of embeddings in F as B. A linear time algorithm to construct such a universal obstruction ˜ B is presented. At the same time, for every type of embeddings of ˜ B, an embedding of B ofthesametypeis determined.
Embedding a Graph Into the Torus in Linear Time
, 1994
"... A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are ..."
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A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...
An algorithm for embedding graphs in the torus
"... An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applyi ..."
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An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1
Projective plane and Möbius band obstructions
, 1997
"... Let S be a compact surface with possibly nonempty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embed ..."
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Let S be a compact surface with possibly nonempty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the projective plane or the Möbius band (for several “canonical” choices of K). Linear time algorithms are presented that either find an embedding extension, or return a “nice” obstruction for the existence of extensions.