Vietoris-Rips Complexes of Split-Decomposable Spaces

Our main object of study is the split-metric decomposition of a finite metric. Given a finite set $X$, a split is a partition $S=\{A,B\}$ of $X$. We denote splits by $A|B$, $B|A$, or $A|\overline{A}$ where $\overline{A}$ is the complement of $A$ in $X$. Given $x\in X$, we denote by $S(x)$ the element of $S$ that contains $x$. A collection ${\mathcal{S}}=\{S\}$ of splits is called a split system. A weighted split system is a pair $({\mathcal{S}},\alpha)$ where $\alpha:{\mathcal{S}}\to{\mathbb{R}}_{>0}$. We write the value of $\alpha$ at $S\in{\mathcal{S}}$ as $\alpha_{S}$.

A split system is weakly compatible if there are no three splits $S_{1},S_{2},S_{3}$ and four elements $x_{0},x_{1},x_{2},x_{3}\in X$ such that

$$S_{j}(x_{i})=S_{j}(x_{0})\text{ if and only if }i=j.$$

Any finite metric space $(X,d)$ has an associated weakly compatible split system (which could be empty) with weights induced by the metric. For any $a_{1},a_{2},b_{1},b_{2}\in X$ (not necessarily distinct), define

$$\beta_{\{a_{1},a_{2}\},\{b_{1},b_{2}\}}:=\max\left\{\begin{array}[]{c}d(a_{1},b_{1})+d(a_{2},b_{2})\\ d(a_{1},b_{2})+d(a_{2},b_{1})\\ d(a_{1},a_{2})+d(b_{1},b_{2})\end{array}\right\}-\big{[}d(a_{1},a_{2})+d(b_{1},b_{2})\big{]}.$$

If $A|B$ is a split of $X$, the isolation index of $A|B$ is the number

$$\alpha_{A|B}:=\min_{\begin{subarray}{c}a_{1},a_{2}\in A\\ b_{1},b_{2}\in B\end{subarray}}\beta_{\{a_{1},a_{2}\},\{b_{1},b_{2}\}}.$$

Note that $\beta_{\{a_{1},a_{2}\},\{b_{1},b_{2}\}}\geq 0$ and, as a consequence, $\alpha_{A|B}\geq 0$. If $\alpha_{A|B}\neq 0$, the split $A|B$ is called a $d$-split. The set

$${\mathcal{S}}(X,d):=\{A|B\text{ split of }X\text{ such that }\alpha_{A|B}\neq 0\}$$

is called the system of $d$-splits of $(X,d)$ and it is weighted by the isolation indices $\alpha_{A|B}$. Bandelt and Dress proved that ${\mathcal{S}}(X,d)$ is always weakly compatible [BD92, Theorem 3]. However, not every metric space $(X,d)$ has $d$-splits. We call such a space split-prime [BD92].

Equation (1) is called the split-metric decomposition of $(X,d)$, or split decomposition for brevity.

Suppose $X$ is a set with $n$ points. Then a weakly compatible split system ${\mathcal{S}}$ on $X$ can have at most $\binom{n}{2}$ splits. The reason is that the vector space of symmetric functions $f:X\times X\to{\mathbb{R}}$ with $0$ diagonal has dimension $\binom{n}{2}$ and the uniqueness in Theorem 2.5 implies that the set of split-metrics $\{\delta_{S}:S\in{\mathcal{S}}\}$ is linearly independent [BD92, Corollary 4].
There are examples of split systems that achieve this bound. Let $V_{n}$ be the vertices of a regular $n$-gon. Any line that passes through two distinct edges $(v_{i-1},v_{i})$ and $(v_{j},v_{j+1})$ separates $V_{n}$ into the sets $\{v_{i},\dots,v_{j}\}$ and $\{v_{j+1},\dots,v_{i-1}\}$. Since there are $n$ edges, this construction gives a split system ${\mathcal{S}}(V_{n})$ with $\binom{n}{2}$ different splits, and it can be verified that ${\mathcal{S}}(V_{n})$ is weakly compatible. It turns out that, up to a bijection of the underlying sets, this is the only example.

The main tool used in [AA17] to compute the homotopy type of $\mathrm{VR}_{r}(\mathbb{S}^{1})$ were the directed cyclic graphs. We give an undirected definition.

We also make the following definition for convenience.

Note that we can give an orientation to our cyclic graphs by orienting an edge $\{a,b\}$ from $a$ to $b$ if $a\preceq b\preceq\sigma(a)$ or $\sigma(b)\preceq a\preceq b$. Thanks to this, the results of [AA17] also hold for undirected cyclic graphs.

We are mainly interested in studying metric spaces whose VR complexes are cyclic.

There is a polytopal complex associated to any weakly compatible split system ${\mathcal{S}}$ on $X$ with weights $\alpha_{S}>0$. Let

$$U({\mathcal{S}}):=\{A\subset X:\text{there exists }S\in{\mathcal{S}}\text{ with }A\in S\}.$$

Given $\phi\in{\mathbb{R}}^{U({\mathcal{S}})}$, let $\mathrm{supp}(\phi):=\{A\in U({\mathcal{S}}):\phi(A)\neq 0\}$. Define

$$H({\mathcal{S}},\alpha):=\{\phi\in{\mathbb{R}}^{U({\mathcal{S}})}:\phi(A)\geq 0\text{ and }\phi(A)+\phi(\overline{A})=\tfrac{1}{2}\alpha(A|\overline{A})\text{ for all }A\in U({\mathcal{S}})\}.$$

Note that $H({\mathcal{S}},\alpha)$ is polytope isomorphic to a hypercube of dimension $|{\mathcal{S}}|$. The Buneman complex of $({\mathcal{S}},\alpha)$ is defined by

$$B({\mathcal{S}},\alpha):=\{\phi\in H({\mathcal{S}},\alpha):A_{1},A_{2}\in\mathrm{supp}(\phi)\text{ and }A_{1}\cup A_{2}=X\Rightarrow A_{1}\cap A_{2}=\emptyset\}.$$

Both $H({\mathcal{S}},\alpha)$ and $B({\mathcal{S}},\alpha)$ are equipped with the $L^{1}$ metric:

$$d_{1}(\phi,\phi^{\prime}):=\sum_{A\in U({\mathcal{S}})}|\phi(A)-\phi^{\prime}(A)|.$$

The space $(B({\mathcal{S}},\alpha),d_{1})$ admits an isometric embedding of $(X,d_{{\mathcal{S}},\alpha})$ via the map $x\mapsto(\phi_{x}:U({\mathcal{S}})\to{\mathbb{R}}_{\geq 0})$ where

$$\phi_{x}(A)=\begin{cases}\frac{1}{2}\alpha_{A|\overline{A}}&\text{if }x\notin A\\ 0&\text{otherwise}.\end{cases}$$

Let $(X,d_{X})$ be a metric space. The tight span of $X$ is the set $T(X,d_{X})\subset{\mathbb{R}}^{X}$ such that for all $f\in T(X,d_{X})$:

1. 1.

   $f(x)+f(y)\geq d_{X}(x,y)$ for all $x,y\in X$;

2. 2.

   $f(x)=\sup_{y\in X}\big{[}d_{X}(x,y)-f(y)\big{]}$ for all $x\in X$.

These properties imply, in particular, that $f(x)\geq 0$ for all $f\in T(X,d_{X})$ and $x\in X$.The tight span is equipped with the $L^{\infty}$ metric

$$d_{\infty}(f,g)=\sup_{x\in X}|f(x)-g(x)|,$$

and there exists an isometric embedding $X\hookrightarrow T(X,d_{X})$ defined by

$$x\mapsto\big{(}h_{x}:y\mapsto d_{X}(x,y)\big{)}.$$

Tight spans are examples of injective metric spaces. A metric space $E$ is injective if for every 1-Lipschitz map $f:X\to E$ and for every isometric embedding $X\hookrightarrow\widetilde{X}$, there exists a 1-Lipschitz extension $\widetilde{f}:\widetilde{X}\to E$ that makes the following diagram commute:

In particular, $T(X,d_{X})$ is the smallest injective space that contains $X$, i.e. $X\hookrightarrow T(X,d_{X})\hookrightarrow E$ for any injective $E$ such that $X\hookrightarrow E$. Furthermore, injective spaces have nice topological and geometric structures.

The tight span of totally decomposable metrics has been studied by several authors throughout the years [BD92, DH01, HKM06, HKM19]. We use the notation and results of [HKM19], one of the most recent papers. Given a weighted split system $({\mathcal{S}},\alpha)$ on $X$, define the map

	$\displaystyle\kappa:{\mathbb{R}}^{U({\mathcal{S}})}$	$\displaystyle\to{\mathbb{R}}^{X}$
	$\displaystyle\phi$	$\displaystyle\mapsto(x\mapsto d_{1}(\phi,\phi_{x})),$

where $d_{1}$ is the $L^{1}$ metric on ${\mathbb{R}}^{U({\mathcal{S}})}$. The first salient property of $\kappa$ is that it sends each element $\phi_{x}$ to $h_{x}$. In other words, $\kappa$ is “constant” on the embedded copy of $X$ inside the Buneman complex and the tight span. The map has much stronger properties when ${\mathcal{S}}$ is weakly compatible.

See the introduction to Section 4 of [HKM06] to see why $\kappa(\phi_{x})=h_{x}$ and why $\kappa$ is 1-Lipschitz. The equivalence between $\kappa(B({\mathcal{S}},\alpha))=T(X,d_{{\mathcal{S}},\alpha})$ and weak compatibility is the main result of [DHM98].

The map $\kappa$ also induces a strong relationship between the polytopal structures of the Buneman complex and the tight span. We need more definitions to state this result.
Let $C$ be a connected polytopal complex. A vertex $v\in C$ is a cut-vertex if $C-\{v\}$ is disconnected. A maximal subcomplex $B\subset C$ that does not have cut-vertices is called a block. The set of blocks of $C$ is denoted as ${\mathcal{B}}(C)$. The block structure of the Buneman complex $B({\mathcal{S}},\alpha)$ can be determined from the properties of ${\mathcal{S}}$ in a straightforward way.