Enumerating simplicial decompositions of surfaces with boundaries

Enumerating simplicial decompositions of surfaces with boundaries

It is well-known that the triangulations of the disc with n + 2 vertices on its boundary are counted by the nth Catalan number C(n) = 1 n+1 ` 2n

Enumerating branched coverings over surfaces with boundaries

The number of nonisomorphic n-fold branched coverings over a given surface with a boundary is determined by the number of nonisomorphic n-fold graph coverings over a suitable bouquet of circles. A similar enumeration can be done for regular branched coverings. Some explicit formulae for enumerations are also obtained. © 2003 Elsevier Ltd. All rights reserved. MSC 2000: 05C10; 05C30; 57M12

Simplicial Decompositions, Tree-decompositions and Graph Minors

The concepts of simplicial decompositions, tree-decompositions and simplicial tree-decompositions were all inspired by a common forerunner: the decompositions of finite graphs used by K. Wagner in his classic paper [ 13 ], in which he proved the equivalence of the 4-Colour-Conjecture to Hadwiger’s Conjecture for n = 5. To show that the 4CC implies Hadwiger’s Conjecture (for n = 5), Wagner used ...

Simplicial Minors and Decompositions of Graphs

Simplicial tree-decompositions of infinite graphs, I

This paper is intended as an introduction to the theory of simplicial decompositions of graphs. It presents, in a unified way, new results as well as some basic old ones (with new proofs). Its main result is a structure theorem for infinite graphs with a simplicial tree-decomposition into primes. The existence and uniqueness of such prime decompositions will be investigated in two subsequent pa...