Arboreal Ultrametrics

TL;DR

The paper extends ultrametric theory to arboreal networks by introducing arboreal ultrametrics, which arise as partial distances represented by ultrametric arboreal networks that can have multiple roots. It proves that any unrooted phylogenetic tree admits a unique ultrametric uprooting and develops an algorithm to construct it, establishing a canonical rooting process. It then characterizes arboreal ultrametrics via a trio of conditions on the associated graph $G_{ ilde{D}}$ and two 3- and 4-point inequalities, linking these objects to symbolic arboreal maps and distance-hereditary graphs. The results yield a rigorous, unique correspondence between arboreal ultrametric partial distances and their ultrametric arboreal representations, with implications for phylogenetics and graph theory, and outline directions for algorithms and generalizations.

Abstract

Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

Figures (5)

• Figure 1: (i) An ultrametric arboreal network with leaf set $X=\{a,b,\dots,f\}$ and (ii) its associated partial distance, where all arcs are directed downwards towards the leaves in $X$. For example, the distance from $a$ to $d$ is 10, which is length of the shortest path from $a$ to $d$ that goes via the root that lies above them both, whereas the distance from $a$ to $f$ is $\infty$ since there is no root that lies above $a$ and $f$.
• Figure 2: (i) The gem is an example of a chordal graph on $5$ vertices, and it is the only chordal forbidden induced subgraphs for Ptolemaic graphs. (ii) The wheel $W_5$ is a non-chordal graph that is obtained from the gem by adding a single edge.
• Figure 3: (i) An arboreal network $N$ on $X=\{a,b,c,d,e,f,g\}$, with three roots $r_1, r_2$ and $r_3$. (ii) The shared ancestry graph $\mathcal{A}(N)$ of $N$. (iii) The underlying graph $\overline N$ of $N$.
• Figure 4: (i) A weighted phylogenetic tree $(T,\lambda)$ that admits the ultrametric network in Figure \ref{['fig:introduction']} as an ultrametric uprooting. (ii) The distance $D_{(T,\lambda)}$.
• Figure 5: (i) A weighted phylogenetic tree $(T,\lambda)$ on $X=\{a,b,c,d\}$. (ii) the output of Algorithm \ref{['alg:ulup']} applied on $(T,\lambda)$ and the wcps $a,b,c,d,e$. (iii), (iv) and (iv) Top, the trees $T|_{\{a,b\}}$, $T|_{\{a,b,c\}}$ and $T|_{\{a,b,c,d\}}$, respectively. Bottom, the intermediate steps of the algorithm. See text for details.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• Theorem 3.5