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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),wheregis the genus of S. This ..."
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Cited by 16 (5 self)
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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),wheregis 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 beafixedsurface. GivenagraphG andanintegerk≥3, we want to find an embedding of G in S of facewidth 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.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... 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 ..."
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Cited by 14 (1 self)
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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 the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
CLIQUE MINORS IN CARTESIAN PRODUCTS OF GRAPHS
, 711
"... ABSTRACT. A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number ηÔGÕis the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G¥H. Our main result is ..."
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Cited by 2 (2 self)
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ABSTRACT. A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number ηÔGÕis the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G¥H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: a planar grid with a vortex of bounded width in the outerface, a cylindrical grid with a vortex of bounded width in each of the two ‘big ’ faces, or a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger’s Conjecture, which states that the chromatic number χÔGÕ�ηÔGÕ. It is open whether Hadwiger’s Conjecture holds for every Cartesian product. We prove that if�VÔHÕ�¡1�χÔGÕ�χÔHÕthen Hadwiger’s Conjecture holds for G¥H. On the other hand, we prove that Hadwiger’s Conjecture holds for all Cartesian products if and only if it holds for all G¥K2. We then show that
Finding Large Clique . . .
, 2009
"... We prove that it is NPcomplete, given a graph G and a parameter h, to determine whether G contains a complete graph Kh as a minor. ..."
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We prove that it is NPcomplete, given a graph G and a parameter h, to determine whether G contains a complete graph Kh as a minor.
EXCLUDING KURATOWSKI GRAPHS AND THEIR DUALS FROM BINARY MATROIDS
, 902
"... Abstract. We consider various applications of our characterization of the internally 4connected binary matroids with no M(K3,3)minor. In particular, we characterize the internally 4connected members of those classes of binary matroids produced by excluding any collection of cycle and bond matroid ..."
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Abstract. We consider various applications of our characterization of the internally 4connected binary matroids with no M(K3,3)minor. In particular, we characterize the internally 4connected members of those classes of binary matroids produced by excluding any collection of cycle and bond matroids of K3,3 and K5, as long as that collection contains either M(K3,3) or M ∗ (K3,3). We also present polynomialtime algorithms for deciding membership of these classes, where the input consists of a matrix with entries from GF(2). In addition we characterize the maximumsized simple binary matroids with no M(K3,3)minor, for any particular rank, and we show that a binary matroid with no M(K3,3)minor has a critical exponent over GF(2) of at most four. 1.
SPECTRAL RADIUS OF FINITE AND INFINITE PLANAR GRAPHS AND OF GRAPHS OF BOUNDED GENUS
, 2009
"... It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2 √ D − 1. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius ρ(G) of a planar graph G of maximum vertex degree D ≥ 4sa ..."
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It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2 √ D − 1. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius ρ(G) of a planar graph G of maximum vertex degree D ≥ 4satisfies √ D ≤ ρ(G) ≤ √ 8D − 16 + 7.75. This result is best possible up to the additive constant—we construct an (infinite) planar graph of maximum degree D, whose spectral radius is √ 8D − 16. This generalizes and improves several previous results and solves an open problem proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus. For every k, these bounds can be improved by excluding K2,k as a subgraph. In particular, the upper bound is strengthened for 5connected graphs. All our results hold for finite as well as for infinite graphs. At the end we enhance the graph decomposition method introduced in the first part of the paper and apply it to tessellations of the hyperbolic