43061363
Counting labelled three-connected and homeomorphically irreducible two-connected graphs
Abstract Recursive procedures are obtained for counting isomorphism classes of three-connected graphs and of two-connected graphs without vertices of degree 2. We apply an enumeration tool developed by R. W. Robinson to count non-isomorphic 2-connected graphs: he expressed the sums of cycle indices of automorphism groups of connected graphs in terms of those of their 2-connected components, and we do the same for 2-connected graphs and their 3-connected components.
14691140
Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs
A graph will be assumed to be finite and unoriented, with no loops or multiple edges; if multiple edges are to be allowed, the term multigraph will be used. A graph or multigraph. will be called k-connected if at least k vertices and their incident edges must be removed to disconnect it (a complete graph is considered to be k-connected for any k). A block (respectively, multiblock) is a 2-connected graph (respectively, multigraph) with at least 2 vertices, and a brick is a 3-connected graph with at least 4 vertices. A Iabelling of a graph or multigraph with n vertices is a l-l correspondence from the set { 1, 2,..., n} onto the set of its vertices. Let A(x, y) be the mixed exponential generating function C,,, An,,,,x”.vmln!, where A,,, is the number of labelled graphs with n vertices and m edges, and let C(x, y) and B(x, y) be analogous generating functions which count labelled connected graphs and labelled blocks, respectively. The following formulae, due to Riddell [lo], appear in one-variable form in [6, pp. 3-111:
28528633
Total Vertex Enumeration in Rooted Planar Maps
8458935
Efficient enumeration of sensed planar maps
We use Liskovets' quotient maps and Robinson's cycle index sums to count 1-, 2- and 3-connected planar maps by number of vertices and edges up to sense-preserving homeomorphism of the embedding sphere. Although Wormald has already counted these maps up to all homeomorphism, sense-reversing as well as sense-preserving, our methods are computationally more efficient for counting these maps up to orientation-preserving homeomorphism and yield closed-form enumeration formulas in the case of 1- and 2-connected maps. Our formula for 1-connected planar maps uses the number of rooted planar maps with i+1 vertices and j+1 faces; we evaluate these numbers using a method that is more efficient than substituting into Tutte's parametric equations, and we also count rooted toroidal maps by number of vertices and faces more efficiently than by substituting into Arques' explicit formula.
15854568
Enumeration of eulerian and unicursal planar maps
Sum-free enumerative formulae are derived for several classes of rooted planar maps with no vertices of odd valency (eulerian maps) and with two vertices of odd valency (unicursal maps). As corollaries we obtain simple formulae for the numbers of unrooted eulerian and unicursal planar maps. Also, we obtain a sum-free formula for the number of rooted bi-eulerian (eulerian and bipartite) maps and some related results.
18489826
Counting unrooted loopless planar maps
We present a formula for the number of n-edge unrooted loopless planar maps considered up to orientation-preserving isomorphism. The only sum contained in this formula is over the divisors of n.
1687075
Using Auxiliary Variables and Implied Constraints to Model Non-Binary Problems
We perform an extensive theoretical and empirical analysis of the use of auxiliary variables and implied constraints in modelling a class of non-binary constraint satisfaction problems called problems of distance. This class of problems include 1-d, 2-d and circular Golomb rulers. We identify a large number of dierent models, both binary and non-binary, and compare theoretically the level of consistency achieved by generalized arc consistency on them. Our experiments show that the introduction of auxiliary variables and implied constraints can signicantly reduce the size of the search space. For instance, our nal models reduce the time to nd an optimal 10-mark Golomb ruler 50-fold.
456544
The Tractability of Global Constraints
Constraint propagation is one of the techniques central to the success of constraint programming. Fast algorithms are used to prune the search space either before or during backtracking search. Propagating global constraints is intractable in general. In this paper, we characterize a number of important questions related to constraint propagation. For example, we consider the two questions: "Is this problem generalized arc-consistent?" and "What are the maximal generalized arc-consistent domains?". We identify dependencies between the tractability and intractability of these questions for finite domain variables. Finally, we prove intractability for a range of global constraints.
16108034
Interleaved and Discrepancy Based Search
We present a detailed experimental comparison of interleaved depth-rst search and depth-bounded discrepancy search, two tree search procedures recently developed with the same goal: to reduce the cost of heuristic mistakes at the top of the tree. Our comparison uses an abstract heur-istic model, and three diierent concrete problem classes: binary constraint satisfaction, quasigroup completion and number partitioning problems. Results indicate that both search strategies often reduce search. In addition, they show that their eeciency depends on a trade-oo between the number of discrepancies (branch points against the heuristic) considered at the top of the tree, and the overhead of expanding branches from these discrepancies. If the number of discrepancies is large, the overhead can outweigh the beneets.
2007911
The Range and Roots Constraints: Specifying Counting and Occurrence Problems
We propose a simple declarative language for specifying a wide range of counting and occurrence constraints. This specification language is executable since it immediately provides a polynomial propagation algorithm. To illustrate the capabilities of this language, we specify a dozen global constraints taken from the literature. We observe one of three outcomes: we achieve generalized arc-consistency; we do not achieve generalized arc-consistency, but achieving generalized arc-consistency is NP-hard; we do not achieve generalized arc-consistency, but specialized propagation algorithms can do so in polynomial time. Experiments demonstrate that this specification language is both efficient and effective in practice.
3264013
Symmetry Breaking Using Value Precedence
We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries.