Distinguished minimal toplogical lassos

TL;DR

This paper investigates the structural properties of minimal topological lassos in dendrograms, characterizing when they uniquely determine the tree topology and introducing the concept of cluster marker maps.

Contribution

It introduces the notion of distinguished minimal topological lassos as claw-free block graphs and characterizes them using cluster marker maps, advancing understanding of dendrogram reconstruction.

Findings

Any minimal topological lasso can be transformed into a distinguished one.

Distinguished lassos are characterized as claw-free block graphs.

Results on heritability of lassos in subtree and supertree problems.

Abstract

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.