44916266
Inversion of Cycle Index Sum Relations for 2- and 3-Connected Graphs
Abstract Algebraic inversion of cycle index sum relations is employed to derive new algorithms for counting unlabeled 2-connected graphs, homeomorphically irreducible 2-connected graphs, and 3-connected graphs. The new algorithms are significantly more efficient than earlier ones, both asymptotically and for modest values of the order. Tables of computed results are included.
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.
28528633
Total Vertex Enumeration in Rooted Planar Maps
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:
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.
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.
1960492
Challenges in securing voice over IP
Although VoIP offers lower cost and greater flexibility, it can also introduce significant risks and vulnerabilities. This article explains the challenges of VoIP security and outlines steps for helping to secure an organization's VoIP network.
2720985
Breaking Row and Column Symmetries in Matrix Models
We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.
3250202
The Range Constraint: Algorithms and Implementation
We recently proposed a simple declarative language for specifying a wide range of counting and occurrence constraints. The language uses just two global primitives: the Range constraint, which computes the range of values used by a set of variables, and the Roots constraint, which computes the variables mapping onto particular values. In order for this specification language to be executable, propagation algorithms for the Range and Roots constraints should be developed. In this paper, we focus on the study of the Range constraint. We propose an efficient algorithm for propagating the Range constraint. We also show that decomposing global counting and occurrence constraints using Range is effective and efficient in practice.
13905371
Combining Symmetry Breaking with Other Constraints: Lexicographic Ordering with Sums
We introduce a new global constraint which combines together the lexicographic ordering constraint with some sum constraints. Lexicographic ordering constraints are frequently used to break symmetry, whilst sum constraints occur in many problems involving capacity or partitioning. Our results show that this global constraint is useful when there is a very large space to explore, such as when the problem is unsatisfiable, or when the search strategy is poor or conflicts with the symmetry breaking constraints. By studying in detail when combining lexicographical ordering with other constraints is useful, we propose a new heuristic for deciding when to combine constraints together.
11418534
Scaling Effects in the CSP Phase Transition
Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify a control parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. This parameter allows different phase transition behaviour to be compared in an uniform manner, for example CSP's generated under different regimes. We then show that within classes, the scaling of behaviour can be modelled by a technique called “finite size scaling”. This applies not only to probability of solubility, as has been observed before in other NP-problems, but also to search cost. Furthermore, the technique applies with equal validity to several different methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much finer grained comparison of algorithms, and for accurate empirical extrapolations of behaviour.