682 research outputs found
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
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An Information-Theoretic Scale for Cultural Rule Systems
Important cultural messages are expressed in nonverbal media such as food, clothing, or the allocation of space or time. For instance, how and what a group of persons eats on a particular occasion may convey public information about that occasion and about the group of persons eating together. Whereas attention seems to be most commonly directed toward the individual character of the information, the present concern is the quantity of public information, as observed in the pattern of nonverbal cultural signs. To measure this quantity, it is proposed that the pattern of cultural signs be encoded as a sequence of abstract symbols (e.g. letters of the alphabet) and its complexity appraised by a suitably adapted form of the measure of Kolmogorov and Chaitin. That is, an algorithmic language is constructed and the mathematical information quantity is reckoned as the length of the shortest program that yields the sequence. In this cultural context, the measure is called "intricacy". By focusing on syntactic structure and pattern variation rather than on background levels, intricacy resists some influences of material wealth that tend to distort comparisons of individuals and groups. A compact mathematical overview of the theory is presented and an experiment to test it within the social medium of food sharing is briefly described
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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
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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
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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. It is proved that the genus distribution of any member of either class is strongly unimodal. 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 cellular orientable imbeddings in the sphere
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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
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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
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