Distinguished minimal toplogical lassos

A classical result in distance based tree-reconstruction characterizes when for a distance D on some finite set X there exist a uniquely determined dendrogram on X (essentially a rooted tree T = (V,E) with leaf set X and no degree two vertices but possibly the root and an edge weighting ω : E → R≥0) such that the distance D(T,ω) induced by (T,ω) on X is D. Moreover, algorithms that quickly reconstruct (T,ω) from D in this case are known. However in many areas where dendrograms are being constructed such as Computational Biology not all distances on X are always available implying that the sought after dendrogram need not be uniquely determined anymore by the available distances with regards to topology of the underlying tree, edge-weighting, or both. To better understand the structural properties a set L ⊆ (X 2 ) has to satisfy to overcome this problem, various types of lassos have been introduced. Here, we focus on the question of when a lasso uniquely determines the topology of a dendrogram’s underlying tree, that is, it is a topological lasso for that tree. We show that any set-inclusion minimal topological lasso for such a tree T can be transformed into a ’distinguished’ minimal topological lasso L for T , that is, the graph (X ,L ) is a claw-free block graph. Furthermore, we characterize such lassos in terms of the novel concept of a cluster marker map for T and present results concerning the heritability of such lassos in the context of the subtree and supertree problems.