The study of decompositions of sets in U, n ^ 1, into disjoint, mutually isometric subsets has a long and distinguished history. In 1928 J. von Neumann [2] proved that an interval in U (with or without endpoints) is decomposable into Ko disjoint sets which are mutually isometric under translations. In 1951 W. Gustin [1] proved that no such decomposition into n, 2 ^ n < No, sets is possible. A b...

We call a vertex x of a graph G = (V,E) a codominated vertex if NG[y] ⊆ NG[x] for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C4, C5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C4, C5, C7)-free well-covered graph, then ver...

Derivation of Cost Function. The projection of the joint probability distribution of the random variables X = (X1, X2, . . . , Xn), associated with the vertices in V , on a decomposable graph G is given by: pG(x) = ∏ C∈C(G) pC(xC) ∏ (C,D)∈T (G) pC∩D(xC∩D) , (1) where x is an instance in the domain of X, which we denote by X . pC(xC) denotes the marginal distribution of random variables belongin...

To a finite metric space (X, d) one can associate the so called tight-span T (d) of d, that is, a canonical metric space (T (d), d∞) into which (X, d) isometrically embeds and which may be thought of as the abstract convex hull of (X, d). Amongst other applications, the tight-span of a finite metric space has been used to decompose and classify finite metrics, to solve instances of the server a...

A continuous quadratic form on a real Banach space X is called decomposable if it is the difference of two nonnegative (i.e., positively semidefinite) continuous quadratic forms. We prove that if X belongs to a certain class of superreflexive Banach spaces, including all Lp(μ) spaces with 2 ≤ p < ∞, then each continuous quadratic form on X is decomposable. On the other hand, on each infinite-di...

Recall that a metric d on a finite set X is called antipodal if there exists a map σ : X → X: x 7→ x so that d(x, x) = d(x, y) + d(y, x) holds for all x, y ∈ X. Antipodal metrics canonically arise as metrics induced on specific weighted graphs, although their abundance becomes clearer in light of the fact that any finite metric space can be isometrically embedded in a more-or-less canonical way...

Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a, b ∈ I and x ∈ V \ I , {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V , and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable...

An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by ...

Let G x H denote the Kronecker product of graphs G and H. Principal results are as follows: (a) If m is even and n 0 (mod 4), then one component of P,.+l x P,+1, and each component of each of CA x Pn+l, Pm+l x (7, and Cm x C, are edge decomposable into cycles of uniform length rs, where r and s are suitable divisors of m and n, respectively, (b) if m and n are both even, then each component of ...

Let G be a finite group. For each integral representation ρ of G we consider ρ−decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with ρ(G)−action, of dimension equal to the degree of ρ, which admit a decomposition of the lattice for X into two G−invariant sublattices isotropic with respect to IH , with one of the sublattices ZG−isomorph...