Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.Comment: 18 pages, 4 figure

Superantigens and nasal polyps

Nasal polyps represent an often severe T-cell-orchestrated eosinophilic upper airway disease with currently unknown pathogenesis, often associated with lower airway disease, such as asthma. Superantigens, predominantly derived from Staphylococcus aureus, are potent activators of T cells, induce the synthesis of IgE in B cells, and have direct effects on pro-inflammatory cells, such as eosinophils. IgE antibodies to S. aureus enterotoxins have been described in polyp tissue, linked to a local polyclonal IgE production and an aggravation of eosinophilic inflammation. Furthermore, such IgE antibodies have also been described in the sera of patients with asthma, and linked to severity of disease and steroid insensitivity. This review summarizes our current understanding of the possible role of S. aureus enterotoxins in chronic severe airway disease, such as nasal polyposis