We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraphK n is decomposable into commuting perfect matchings if and only if n is a 2-power. Also, it is shown that the complete graph Kn is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 200...

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...

Wediscuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable, that is, whether it can be partitioned into connected parts of orders α1, α2, . . . , αk for every α1, α2, . . . , αk summing up to the order of the graph. In contrast, we show that the decision problem...

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...

In this article, we deal with graphs modelling interconnection networks of parallel systems (parallel computers, networks of workstations,..). We want to share the nodes of such a network between many users, each one needing a given number of nodes. Thus, a graph G with N vertices is said to be decomposable if for each set fn 1 We show that determining if a given tripode (three disjoint chains ...

Recently the primal-dual method of multipliers (PDMM), a novel distributed optimization method, was proposed for solving a general class of decomposable convex optimizations over graphic models. In this work, we first study the convergence properties of PDMM for decomposable quadratic optimizations over tree-structured graphs. We show that with proper parameter selection, PDMM converges to its ...

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...

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1, ..., nk) of positive integers such that n1 + ... + nk = n there exists a partition (V1, ..., Vk) of the vertex set of G such that for each i ∈ {1, ..., k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence nu...