Abstract
The set of all orientable cellular embeddings of a graph has the intrinsic structure of adjacency between embeddings based on elementary operations on rotation schemes. Several types of elementary operations were considered in the past, usually in proofs of interpolation theorems: moving a single arc within its local rotation, moving both ends of an edge in the respective local rotations, and interchanging two arcs in the local rotation at a given vertex, see [2,6,7,10]. We call these operations rotation moves. Each type of a rotation move gives rise to the structure of a stratified graph on the set of all embeddings of a given graph. Stratified graphs were studied by Gross, Rieper, and Tucker [5,6,8], although they were implicit already in the works of Duke [2] and Stahl [10]. Very little is known about stratified graphs in general, although their structure is crucial for understanding the entire system of all embeddings of a given graph. In the present paper we focus on embeddings that correspond to local maxima in stratified graphs. We call a cellular embedding of a graph into an orientable surface locally maximal if its genus cannot be raised by moving a single arc within its local rotation. Somewhat surprisingly, the concept of a locally-maximal embedding does not depend on which type of a move is taken as a basis for the stratified graph, indicating its important position in the hierarchy of graph embeddings between the minimum genus and the maximum genus.
This research was partially supported by APW-0223-10, VEGA 1/1005/12, UK 513/2013, and by the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation, under the contract APW-ESF-EC-0009-10.
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Kotrbčík, M., Škoviera, M. (2013). Locally-maximal embeddings of graphs in orientable surfaces. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_35
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DOI: https://doi.org/10.1007/978-88-7642-475-5_35
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