Let F ={H1, . . . ,Hk} be a family of graphs. A graph G is called totally F -decomposable iffor every linear combination of the form α1e(H1) + · · · + αke(Hk) = e(G) where each αi is anonnegative integer, there is a coloring of the edges of G with α1 + · · · + αk colors such thatexactly αi color classes induce each a copy of Hi, for i = 1, . . . , k. We prove that if F i...

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamiltondecomposable. Liu has shown that for |A| even, if S = {s1, . . . , sk} ⊂ A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A;S?), is decomposable into k Hamilton cycles, where S? denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regu...

We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, defined as wellconnected subsets of vertices. The decomposition is based on a statistical description which encourages clusters to be well connected and few...

We make several contributions in accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard...

Modern approaches for optimally learning Bayesian network structures require decomposable scores. Such approaches include those based on dynamic programming and heuristic search methods. These approaches operate in a search space called the order graph, which has been investigated extensively in recent years. In this paper, we break from this tradition, and show that one can effectively learn s...

Decomposable triangulated graphs have been shown to be efficient and effective for modeling the probabilistic spatio-temporal structure of brief stretches of human motion. In previous work such model structure was handcrafted by expert human observers and labeled data were needed for parameter learning. We present a method to build automatically the structure of the decomposable triangulated gr...

In geometric constraint solving, well constrained geometric problems can be abstracted as Laman graphs. If the graph is tree decomposable, the constraint-based geometric problem can be solved by a Decomposition-Recombination planner based solver. In general decomposition and recombination steps can be completed only when other steps have already been completed. This fact naturally defines a hie...

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E1∪E2∪ ...∪Ek of the edge set E(G) such that each Ei induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into 3 locally irre...

Let F ={H1, . . . ,Hk} be a family of graphs. A graph G with m edges is called totallyF -decomposable if for every linear combination of the form α1e(H1)+ · · ·+αke(Hk) = m whereeach αi is a nonnegative integer, there is a coloring of the edges of G with α1 + · · ·+ αk colorssuch that exactly αi color classes induce each a copy of Hi, for i = 1, . . . , k. We prove that ...

A graph G = (V,E) is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to |V |, there is a sequence of vertex-disjoint subsets of V whose orders are given by τ , and which induce connected graphs. The aim of this paper is to study the recursive version of this problem on a special class of graphs called suns. This paper is a complement of [O. Baudon, F. Gilber...