Obstructions of Connectivity Two for Embedding Graphs into the Torus (original) (raw)

An approach for torus embedding

Proceedings of the 1999 ICPP Workshops on Collaboration and Mobile Computing (CMC'99). Group Communications (IWGC). Internet '99 (IWI'99). Industrial Applications on Network Computing (INDAP). Multimedia Network Systems (MMNS). Security (IWSEC). Parall

A graph is embeddable on a surface S if it can be drawn on S with no crossing edges. A topological obstruction for a surface S is a graph G that does not embed on S, but for all edges e in G, G − e embeds on S. A minor order obstruction has the additional property that, for all edges e, G • e (G contract e) also embeds on S. Solving the well-studied problem of finding a complete set of obstructions for the torus is facilitated by having a large database of torus obstructions. With this in mind, we have designed a new exponential torus embedding algorithm inspired by Demoucron's O(n 2) planar embedding algorithm. Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented. Our implementation of our new algorithm is faster than implementations of previous exponential algorithms that have been used to find torus obstructions.

On the Connectivity of Graphs Embedded in Surfaces

Journal of Combinatorial Theory, Series B, 1998

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus #Ä >0, #Ä {2) there is a complete graph of orientable genus # (respectively, nonorientable genus #Ä ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus #Ä ), for every # (respectively #Ä ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, #Ä >0, #Ä {2) there is a complete graph K n which embeds in S # (respectively N #Ä ) such that n is the chromatic number of surface S # (respectively, the chromatic number of surface N #Ä ). One then easily observes that the connectivity of this K n attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity } max of the orientable surface S # to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity } gen (S # ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds } max (N #Ä ) and } gen (N #Ä ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity } max (S # ) or } max (N #Ä ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(K n )+1, #(K n+1 )] and [#Ä (K n )+1, #Ä (K n+1 )] with respect to their genus and maximum connectivities.

Obstructions for two-vertex alternating embeddings of graphs in surfaces

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

Embeddings of Small Graphs on the Torus

2003

Embeddings of graphs on the torus are studied. All 2-cell embeddings of the vertex-transitive graphs on 12 vertices or less are constructed. Their automorphism groups and dual maps are also constructed. A table of em- beddings is presented. 1. Toroidal Graphs

A Large Set of Torus Obstructions and How They Were Discovered

The Electronic Journal of Combinatorics, 2018

We outline the progress made so far on the search for the complete set of torus obstructions and also consider practical algorithms for torus embedding and their implementations. We present the set of obstructions that are known to-date and give a brief history of how these graphs were found. We also describe a nice algorithm for embedding graphs on the torus which we used to verify previous results and add to the set of torus obstructions. Although it is still exponential in the order of the graph, the algorithm presented here is relatively simple to describe and implement and fast-in-practice for small graphs.It parallels the popular quadratic planar embedding algorithm of Demoucron, Malgrange, and Pertuiset.

An obstruction to embedding graphs in surfaces

Discrete Mathematics, 1989

It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.

Drawing a disconnected graph on the torus (Extended abstract)

Electronic Notes in Discrete Mathematics, 2015

We study drawings of graphs on the torus with crossings allowed. A question posed in [4], specialized to the case of the torus, asks, whether for every disconnected graph there is a drawing in the torus with the minimal number of crossings, such that one of the graphs is drawn in a planar disc. We reduce the problem to an interesting question from the geometry of numbers and solve a special case.

Embedding graphs in the torus in linear time

Lecture Notes in Computer Science, 1995

A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph 12 of G of small branch size that cannot be embedded in the torus.