Abstract. Let K = C ∪ e1 ∪ e2 be a subgraph of G, consisting of a cycle C and disjoint paths e1 and e2, connecting two interlacing pairs of vertices in C. Suppose that K is embedded in the MöbiusbandinsuchawaythatC lies on its boundary. An algorithm is presented which in linear time extends the embedding of K to an embedding of G, if such an extension is possible, or finds a “nice ” obstruction for such embedding extensions. The structure of obtained obstructions is also analysed in details. Key words. surface embedding, obstruction, Möbius band, algorithm AMS subject classifications. 05C10, 05C85, 68Q20 1. Introduction. Let K be a subgraph of a graph G. A K-bridge (or a Kcomponent)inG is a subgraph of G which is either an edge e ∈ E(G)\E(K) (together with its endpoints) which has both endpoints in K, or it is a connected component of G − V (K) together with all edges (and their endpoints) between this component and K. EachedgeofaK-bridge B having an endpoint in K is a foot of B. The vertices

We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.

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