In this paper we introduce a new graph composition, called 2-amalgam, and we prove that the 2-amalgam of perfect graphs is perfect. This composition generalizes many of the operations known to preserve perfection, such as the clique identification, substitution, join and amalgam operations. We give polynominal-time algorithms to determined whether a general graph is decomposable with respect to...

The last decade has witnessed the apparition of applied problems typified by very high-dimensional variables, in marketing database or gene expression studies for instance. Graphical models (Lauritzen (1996)) enable concise representations of associational relations between variables. If the graph is known, the parameters of the model are easily estimated. However, a quite challenging issue is ...

The incidence coloring game has been introduced in [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980– 1987] as a variation of the ordinary coloring game. The incidence game chromatic number ιg(G) of a graph G is the minimum number of colors for which Alice has a winning strategy when playing the incidence coloring game on G. In [C. Charpentier and É. Sopen...

R. Häggkvist proved that every 3-regular bipartite graph of order 2n with no component isomorphic to the Heawood graph decomposes the complete bipartite graph K6n,6n. In [1] Cichacz and Froncek established a necessary and sufficient condition for the existence of a factorization of the complete bipartite graph Kn,n into generalized prisms of order 2n. In [2] and [3] Cichacz, Froncek, and Kovar ...

The graph G(N; d) has vertex set V = {0; 1; : : : ; N − 1}, with {v; w} an edge if v − w ≡ ±di(modN ) for some 06i6dlogd Ne − 1. We show that the circulant graph G(cd; d) is Hamilton decomposable for all positive integers c; d, and m with c¡d. This extends work of Micheneau and answers a special case of a question of Alspach. c © 2000 Elsevier Science B.V. All rights reserved.

We derive methods for enumerating the distinct junction tree representations for any given decomposable graph. We discuss the relevance of the method to estimating conditional independence graphs of graphical models and give an algorithm that, given a junction tree, will generate uniformly at random a tree from the set of those that represent the same graph. Programs implementing these methods ...

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of mu...

Let H be a graph, and let CH(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating CH(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorp...

We present VLSI algorithms that run in O(n 2) steps for nding (and reporting) a fundamental set of cycles and a fundamental set of cutsets of an undirected graph on an nxn mesh of processors (SIMD). Both algorithms are decomposable and run in O(n 4 =k 2) steps for a graph with n vertices when the size of the mesh-connected computer is kxk and k < n. An idea similar to the one used in nding a fu...

We present VLSI algorithms that run in O(n 2) steps for nding (and reporting) a fundamental set of cycles and a fundamental set of cutsets of an undirected graph on an nxn mesh of processors (SIMD). Both algorithms are decomposable and run in O(n 4 =k 2) steps for a graph with n vertices when the size of the mesh-connected computer is kxk and k < n. An idea similar to the one used in nding a fu...