Tree-Robinson Dissimilarities

Abstract

Given a dissimilarity \(\textbf{d}\) on a n-set X, a tree T with vertex set X is said to be R-compatible with \((X, \textbf{d})\) if for all \( x, z \in X\) and y on the path (in T) between x and z, we have \(\textbf{d}(x, z) \ge \max \{\textbf{d}(x, y),\) \(\textbf{d}(y, z)\)}. If T is R-compatible with \((X, \textbf{d})\), then T is a minimum spanning of \((X, \textbf{d})\). We say that \((X, \textbf{d})\) is tree-Robinson if all its minimum spanning trees are R-compatible. In this paper, we give a local characterization of these dissimilarities, in the sense that, although the definition of tree-Robinson dissimilarities involves all minimum spanning trees of \((X, \textbf{d})\), our characterization involves only some of them. This yields an efficient \(O(n^3)\) algorithm to recognize these dissimilarities.