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h1. Introduction

The purpose of Rich Newick strings is to succinctly represent the main features of phylogenetic networks in a character encoded human readable form. The format is inspired primarily by the Newick format for phylogenetic tree representation and the Extended Newick format for phylogenetic network representation as proposed by Cardona et al \[1\]. 

For the purposes of Rich Newick we define a phylogenetic network to be either:

* a unrooted phylogenetic X-tree (see \[2\], p.17)

or

* a connected phylogenetic X-network (defined below)

used to model evolutionary relationships.

We define a connected phylogenetic X-network N to be an ordered pair (G, f), where:

{latex}
\begin{itemize}
\item $G = (V, E)$ is a connected, directed, acyclic graph (connected DAG) with $V = \{r\}, \bigcup V_{L} \bigcup V_{T} \bigcup V_{N}$, where:
\begin{itemize}
\item indeg(r) = 0 (r is the root of N); \\*
\item for all $v$ an element of $V_{L}$, indeg(v) = 1 and outdeg(v) = 0 ($V_{L}$ are the leaves of N); \\*
\item for all $v$ an element of $V_{T}$, indeg(v) = 1 and outdeg(v) $\ge$ 1 ($V_{T}$ are the tree-nodes of N); and, \\*
\item for all $v$ an element of $V_{h}$, indeg(v) $\ge$ 2 and outdeg(v) = 1 ($V_{h}$ are the hybrid nodes of N). \\*
\end{itemize}
\item $f : V_{L} \rightarrow X$ is the leaf-labeling function, which is a bijection from $V_{L}$ to X.
\end{itemize}
{latex}

h1. Representing Phylogenetic X-Networks 

Consider the following phylogenetic X-network with X = \{ 1, 2 \}: