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For a graph G, let 9(G) be the family of strong orientations of G, d(G) = min{d(D) / D t 9' (G)} and p(G) = d(G)-d(G), where d(G) and d(D) are the diameters of G and D respectively. In this paper we show that p(G) = 0 if G is a Cartesian product of (I) paths, and (2) paths and cycles, which satisfy some mild conditions.

A vertex set X of a digraph D = (V, A) is a kernel if X is independent (i.e., all pairs of distinct vertices of X are non-adjacent) and for every v ∈ V − X there exists x ∈ X such that vx ∈ A. A vertex set X of a digraph D = (V, A) is a quasi-kernel if X is independent and for every v ∈ V − X there exist w ∈ V − X, x ∈ X such that either vx ∈ A or vw, wx ∈… (More)

In this paper, we present a formula for computing the Tutte polynomial of the signed graph formed from a labeled graph by edge replacements in terms of the chain polynomial of the labeled graph. Then we define a family of 'ring of tangles' links and consider zeros of their Jones polynomials. By applying the formula obtained, Beraha-Kahane-Weiss's theorem… (More)