No abstract

We find a set of necessary and sufficient conditions under which the weight w : E → R + on the graph G = (V, E) can be extended to a pseudometric d : V × V → R + . If these conditions hold and G is a connected graph, then the set Mw of all such extensions is nonvoid and the shortest-path pseudometric dw is the greatest element of Mw with respect to the partial ordering d1 d2 if and only if d1 (u, v) d2(u, v) for all u, v ∈ V . It is shown that every nonvoid poset (Mw, ) contains the least element ρ0,w if and only if G is a complete k-partite graph with k 2 and in this case the explicit formula for computation of ρ0,w is obtained.

Abstract. Let (X, d) be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality |Sp(X)| |X| where Sp(X) = {d(x, y) : x, y ∈ X}. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric (X, d) satisfying the equality | Sp(X)| = |X| for given Sp(X). Moreover it is shown that every finite semimetric space Z is an image under a composition of mappings f : X → Y and g : Y → Z such that X and Y are finite ultrametric space, X satisfies the above equality, f is an ε-isometry with an arbitrary ε > 0, and g is a ball-preserving map.

Let (X, d X ) and (Y, d Y ) be semimetric spaces with distance sets D(X) and, respectively, D(Y ). A mapping F : X → Y is a weak similarity if it is surjective and there exists a strictly increas-It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F : X → Y is an isometry if X and Y are ultrametric and compact with D(X) = D(Y ). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.