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Abstract not found

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 non-contractible and a shortest non-separating cycle of G. If k is an integer, we can compute such a non-trivial 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 edge-width or face-width of a graph is bounded from above by a constant. This also implies an output-sensitive algorithm to compute a shortest non-trivial cycle that runs in O(gnk) time, where k is the length of the cycle.

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...

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 face-width at most two. 1 Introduction We follow standard graph theory terminology as used, for example, in [1]. Let \Pi be a (2-cell) embedding of a graph G into a nonplanar surface S, i.e. a closed surface distinct from the 2-sphere. Then we define the face-width 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 face-width at most two. Robertson and Vitra...

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

Let S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every onesided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width 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.

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 trade-off 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 doubly-toroidal 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 non-orientable crossing sequence ~ c 0 ; ~c 1 ; ~c 2 ; : : : for drawings on non-orientable surfaces. We show that for every ~ c 0 ? ~ c 1 ? 0 there exists a graph with non-orientable crossing sequence ~ c 0 ; ~c 1 ; 0. We conjecture that every strictly-decreasing sequence...