Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.

We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6. 1

Abstract. The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of these irreducible triangulations. 1.

Given a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides view-dependent refinement and facilitates the computation of parameterization. For a triangulated closed surface of n vertices and genus g, we prove that there is a constant c ? 0 such that if n ? c \Delta g, then a greedy strategy can identify \Theta(n) topology-preserving edge contractions that do not interfere with each other. Further, they affect only a constant number of triangles. Repeatedly identifying and contracting such edges produces a topology-preserving hierarchy of O(n + g ) size and O(logn + g) depth. The genus g is very small when compared with n for large models in practice. The identification of edges can be enhanced by selecting edges based on the error of their contractions as measured by some known heuristics.

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R 3 for all vertex-minimal triangulations of the orientable surfaces of genus g = 3 and g = 4. Moreover, for the first time, examples of simplicial polyhedra in R 3 of genus 5 with 12 vertices were obtained. 1

We present methods to automatically compute nice drawings and embeddings of combinatorial objects. As in previous approaches [33, 41] this can be achieved by using a physical model based on spring forces [32] or on electrical forces [10, 21]. This article generalizes these earlier results, since we use a physical model where the edges in a combinatorial complex are represented as springs and a certain number of the vertices has fixed positions or carries electric charges. Also non-linear spring forces are considered and --- in contrast to the previous 2-dimensional models --- we treat algorithmic solutions for arbitrary dimension d. The minimization of the energy of this system yields in most cases automatically nice drawings and embeddings of combinatorial objects. To have a measure of niceness we define ratios of extreme volumes, e.g., the ratio of the shortest to the longest edge. For some combinatorial objects it is necessary to find a compatible object which has additional structu...

Given an algebraic hypersurface O in R d, how many simplices are necessary for a simplicial complex isotopic to O? We address this problem and the variant where all vertices of the complex must lie on O. We give asymptotically tight worst-case bounds for algebraic plane curves of degree n. Our results gradually improve known bounds in higher dimensions; however, the question for tight bounds remains unsolved for d ≥ 3.

This paper studies the following question: Given a surface Σ and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n − ω) + 2 ω and 8n + 3 2 2ω + o(2 ω), where ω is the maximum integer such that the complete graph Kω embeds in Σ. For the surfaces S0, S1, S2, N1, N2, N3 and N4 we establish an exact answer.

Given a surface triangulation T of and a subset X of its vertex set V (T), we define a restricted edge contraction as a contraction of an edge connecting X and V (T)−X. Boundary vertices in V (T) − X are only allowed to be contracted to the boundary vertices in X adjacent through boundary edges. In this paper, we prove that if a triangulation T of the sphere with boundary satisfies some connectivity conditions, then all the vertices in V (T) − X can be merged into X by restricted edge contractions. We also prove that the similar properties hold for a triangulation of the sphere with features. 1