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The crossing number of a projective graph is quadratic in the facewidth
 ELECTRON J. COMBIN
, 2008
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OutputSensitive Algorithm for the EdgeWidth of an Embedded Graph
, 2010
"... Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle w ..."
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Cited by 6 (1 self)
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Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edgewidth or facewidth of a graph is bounded from above by a constant. This also implies an outputsensitive algorithm to compute a shortest nontrivial cycle that runs in O(gnk) time, where k is the length of the cycle.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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Cited by 3 (0 self)
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...
FourTerminal Reducibility and ProjectivePlanar WyeDeltaWyeReducible Graphs
 J. GRAPH THEORY
, 2000
"... A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible w ..."
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Cited by 2 (0 self)
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A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible when at least three of the vertices lie on the same face. Using this result we characterize Y∆Yreducible projectiveplanar graphs. We also consider terminals in projectiveplanar graphs, and establish that graphs of crossingnumber one are Y∆Yreducible.
Blocking nonorientability of a surface
, 2001
"... Let � be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in � is a blockage if every onesided simple closed curve in � crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in � with sufficiently large ..."
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Let � be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in � is a blockage if every onesided simple closed curve in � crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in � with sufficiently large facewidth is “roughly” equal to one half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar [7] and replaced by a similar one. In this paper, it is proved that the conjectures in [7, 9] hold up to a constant error term: For any graph G embedded in �, the orientable genus of G differs from the conjectured value at most by O(g 2), where g is the genus of �.
Apex Graphs With Embeddings of FaceWidth Three
, 1994
"... Aa apex graph is a graph which has a vertex whose removal makes the resulting graph planar. Embeddings of apex graphs having facewidth three are characterized. Surprisingly, there are such embeddings of arbitrarily large genus. This solves a problem of Robertson and Vitray. We also give an eleme ..."
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Cited by 1 (0 self)
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Aa apex graph is a graph which has a vertex whose removal makes the resulting graph planar. Embeddings of apex graphs having facewidth three are characterized. Surprisingly, there are such embeddings of arbitrarily large genus. This solves a problem of Robertson and Vitray. We also give an elementary proof of a result of Robertson, Seymour, and Thomas [5] that any embedding of an apex graph in a nonorientable surface has facewidth at most two. 1 Introduction We follow standard graph theory terminology as used, for example, in [1]. Let \Pi be a (2cell) embedding of a graph G into a nonplanar surface S, i.e. a closed surface distinct from the 2sphere. Then we define the facewidth fw(\Pi) (also called the representativity) of the embedding \Pi as the smallest number of (closed) faces of G in S whose union contains a noncontractible curve. It is not difficult to see (cf. [6, 7]) that a planar graph embedded in a nonplanar surface has facewidth at most two. Robertson and Vitra...
Topological Graph Theory  A Survey
 Cong. Num
, 1996
"... this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1 ..."
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this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1
How to Eliminate Crossings by Adding Handles or Crosscaps
"... Let c k = cr k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c 0 ; c 1 ; c 2 ; : : : encodes the tradeoff between adding handles and decreasing crossings. We focus on sequences of the type c 0 ? c ..."
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Let c k = cr k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c 0 ; c 1 ; c 2 ; : : : encodes the tradeoff between adding handles and decreasing crossings. We focus on sequences of the type c 0 ? c 1 ? c 2 = 0; equivalently, we study the planar and toroidal crossing number of doublytoroidal graphs. For every ffl ? 0 we construct graphs whose orientable crossing sequence satisifies c 1 =c 0 ? 5=6 \Gamma ffl. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save 5 times more crossings. We similarly define the nonorientable crossing sequence ~ c 0 ; ~c 1 ; ~c 2 ; : : : for drawings on nonorientable surfaces. We show that for every ~ c 0 ? ~ c 1 ? 0 there exists a graph with nonorientable crossing sequence ~ c 0 ; ~c 1 ; 0. We conjecture that every strictlydecreasing sequence...
Algorithms for the EdgeWidth of an Embedded Graph ∗
, 2010
"... Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle w ..."
Abstract
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Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edgewidth or facewidth of a graph is bounded from above by a constant. This also implies an outputsensitive algorithm to compute a shortest nontrivial cycle that runs in O(gnk) time, where k is the length of the cycle. We also give an approximation algorithm for the shortest nontrivial cycle. If a parameter 0 < ε < 1 is given, we compute in O(gn/ε) time a nontrivial cycle whose length is at most 1+ε times the length of the shortest nontrivial cycle.