Combinatorial maps

Combinatorial maps

The classical approach to maps, as surveyed by Coxeter and Moser (“Generators and Relations for Discrete Groups,” Springer-Verlag, 1980), is by cell decomposition of a surface. A more recent approach, by way of graph embedding schemes, is taken by Edmonds (Notices Amer. Math. Sot. 7 (1960), 646), Tutte (Canad. J. Math. 31 (5) (1979), 986-1004), and others. Our intention is to formulate a purely...

Introduction to Combinatorial Maps

Preface Maps as a mathematical main topic arose probably from the four color problem[Ore1] and the more general map coloring problem[Rin1] in the mid of the nineteenth century. I could not list even main references on them because it is well known for mathematicians and beyond the scope of this lecture notes. Here, I only intend to present a comprehensive theory of combinatorial maps as a rigor...

Signatures of Combinatorial Maps

In this paper, we address the problem of computing a canonical representation of an n-dimensional combinatorial map. To do so, we define two combinatorial map signatures: the first one has a quadratic space complexity and may be used to decide of isomorphism with a new map in linear time whereas the second one has a linear space complexity and may be used to decide of isomorphism in quadratic t...

Pyramids with Combinatorial Maps

Combinatorial Maps with Normalized Knot

We consider combinatorial maps with fixed combinatorial knot numbered with augmenting numeration called normalized knot. We show that knot’s normalization doesn’t affect combinatorial map what concerns its generality. Knot’s normalization leads to more concise numeration of corners in maps, e.g., odd or even corners allow easy to follow distinguished cycles in map caused by the fixation of the ...