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The computational complexity of constructing the imbeddings of a given graph into surfaces of different genus is not well understood. In this paper, topological methods and a reduction to linear matroid parity are used to develop a polynomial-time algorithm to find a maximum-genus cellular imbedding. This seems to be the first imbedding algorithm for which(More)
The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. It is proved that the genus distribution of any member of either class is strongly unimodal. These are the first two(More)
It is demonstrated that a given value of average genus is shared by at most nitely many 2-connected simplicial graphs and by at most nitely many 3-connected graphs. Moreover, the distribution of values of average genus is sparse, in the following sense: within any nite real interval, there are at most nitely many different numbers that are values of average(More)
Graphs of small average genus are characterized. In particular, a Kuratowski-type theorem is obtained: except for nitely many graphs, a cutedge-free graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowski-type theorem for graphs of maximum genus 1 is also given. Some of(More)
Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined. and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number one is a limit point of the set of possible values for(More)