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...

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...

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...

We study Markov bases of decomposable graphical models consisting of primitive moves (i.e. square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph...

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We al...

Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation poly...