A Class of Unrooted Phylogenetic Networks Inspired by the Properties of Rooted Tree-Child Networks

TL;DR

New classes called q-cuttable networks are proposed, which have many of the desirable properties, similar to tree-child networks in the rooted case, including being recognizable in polynomial time, for all q\geq 1.

Abstract

A directed phylogenetic network is tree-child if every non-leaf vertex has a child that is not a reticulation. As a class of directed phylogenetic networks, tree-child networks are very useful from a computational perspective. For example, several computationally difficult problems in phylogenetics become tractable when restricted to tree-child networks. At the same time, the class itself is rich enough to contain quite complex networks. Furthermore, checking whether a directed network is tree-child can be done in polynomial time. In this paper, we seek a class of undirected phylogenetic networks that is rich and computationally useful in a similar way to the class tree-child directed networks. A natural class to consider for this role is the class of tree-child-orientable networks which contains all those undirected phylogenetic networks whose edges can be oriented to create a tree-child network. However, we show here that recognizing such networks is NP-hard, even for binary networks, and as such this class is inappropriate for this role. Towards finding a class of undirected networks that fills a similar role to directed tree-child networks, we propose new classes called $q$-cuttable networks, for any integer $q\geq 1$. We show that these classes have many of the desirable properties, similar to tree-child networks in the rooted case, including being recognizable in polynomial time, for all $q\geq 1$. Towards showing the computational usefulness of the class, we show that the NP-hard problem Tree Containment is polynomial-time solvable when restricted to $q$-cuttable networks with $q\geq 3$.

Figures (14)

• Figure 1: An unrooted binary phylogenetic network $U$ on five leaves $\{a,b,c,d,e\}$ that is $2$-cuttable but not $3$-cuttable (left) and a rooted binary phylogenetic network $N$ that is a tree-child orientation of $U$ (right). Observe that $N$ can be obtained from $U$ by subdividing the edge $\{u,v\}$ with a new vertex $\rho$ and directing each edge in the resulting graph.
• Figure 2: The connection gadget (left) and the reticulation gadget (right).
• Figure 3: A rooted phylogenetic network $N$ that has the connection gadget as a subgraph. If $N$ is tree-child, then the edges of the connection gadget can, for example, be directed in one of the two ways shown. Dotted rectangles indicate omitted parts of $N$.
• Figure 4: A rooted phylogenetic network $N$ that has the reticulation gadget as a subgraph. If $N$ is tree-child, the edges of the reticulation gadget are directed in exactly one of the two ways shown. Dotted rectangles indicate omitted parts of $N$.
• Figure 5: The unrooted phylogenetic network $U_\Phi$ for $\Phi=(l_1^1\vee l_1^2\vee l_1^3)\wedge(l_2^1\vee l_2^2\vee l_2^3)\wedge (l_3^1\vee l_3^2\vee l_3^3)\wedge(l_4^1\vee l_4^2\vee l_4^3)=(x\vee \bar{y}\vee z)\wedge(\bar{x}\vee y\vee \bar{z})\wedge(x\vee y\vee\bar{z})\wedge(\bar{x}\vee\bar{y}\vee z)$. To simplify the presentation, some leaf labels are omitted. The $s$-terminal (resp. $t$-terminal) of each connection and reticulation gadget is indicated in green (resp. blue). Note that $p_1$ is the $t$-terminal of $R_r$ and the $s$-terminal of $R^1$.

Theorems & Definitions (39)

• Lemma 1
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Theorem 6
• Proposition 7