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 $\omega:E\to \mathbb R_{\geq 0}$) such that the distance $D_{(T,\omega)}$ induced by $(T,\omega)$ on $X$ is $D$. Moreover, algorithms that quickly reconstruct $(T,\omega)$ 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 $\cL\subseteq {X\choose 2}$ 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 $\cL$ for $T$, that is, the graph $(X,\cL)$ 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.