On the Average Genus of a Graph

Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number one is a limit point of the set of possible values for average genus and that the complete graph K4 is the only 3-connected graph whose average genus is less than one. Several problems for future study are suggested

Stratified Graphs

Two imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, the collection of imbeddings of G may be regarded as a "stratified graph", denoted SG. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the "kth stratum", and one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is proved that the stratified graph is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three and that the spanning subgraph of SG corresponding to moving only one edge-end is a cartesian product of graphs whose underlying isomorphism types depend only on the valence sequence for G

Stratified graphs for imbedding systems

AbstractTwo imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, a collection of imbeddings S of G, called a ‘system’, may be represented as a ‘stratified graph’, and denoted SG; the focus here is the case in which S is the collection of all orientable imbeddings. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the ‘kth stratum’, and the cardinality of that set of imbeddings is called the ‘stratum size’; one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is known that the genus distribution is not a complete invariant, even when the category of graphs is restricted to be simplicial and 3-connected. However, it is proved herein that the link of each point — that is, the subgraph induced by its neighbors — of SG is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three. This supports the plausibility of a probabilistic approach to graph isomorphism testing by sampling higher-order imbedding distribution data. A detailed structural analysis of stratified graphs is presented

Genus Distributions for Two Classes of Graphs

The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their polygonal orientable imbeddings in the sphere

Graph Imbeddings and Overlap Matrices (Preliminary Report)

Mohar has shown an interesting relationship between graph imbeddings and certain boolean matrices. In this paper, we show some interesting properties of this kind of matrices. Using these properties, we give the distributions of nonorietable imbeddings of several interesting infinite families of graphs, including cobblestone paths, closed-end ladders for which the distributions of orientable imbeddings are known

Total embedding distributions of Ringel ladders

The total embedding distributions of a graph is consisted of the orientable embeddings and non- orientable embeddings and have been know for few classes of graphs. The genus distribution of Ringel ladders is determined in [Discrete Mathematics 216 (2000) 235-252] by E.H. Tesar. In this paper, the explicit formula for non-orientable embeddings of Ringel ladders is obtained

Genus Distributions of cubic series-parallel graphs

We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.Comment: 21 page

Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph

The maximum genus gamma_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least gamma_M(G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong\u27s combinatorial characterisation of maximum genus