A Wild Sheep Chase Through an Orchard

TL;DR

It is proved that deciding whether an undirected, binary phylogenetic network is an orchard -- or equivalently, whether it has an orientation that makes it a rooted orchard -- is NP-hard.

Abstract

Orchards are a biologically relevant class of phylogenetic networks as they can describe treelike evolutionary histories augmented with horizontal transfer events. Moreover, the class has attractive mathematical characterizations that can be exploited algorithmically. On the other hand, undirected orchard networks have hardly been studied yet. Here, we prove that deciding whether an undirected, binary phylogenetic network is an orchard -- or equivalently, whether it has an orientation that makes it a rooted orchard -- is NP-hard. For this, we introduce a new characterization of undirected orchards which could be useful for proving positive results.

Figures (13)

• Figure 1: A directed orchard $N_d$, an undirected orchard $N_u$, and a rooted tree $T$. $N_d$ is an orchard since it can be obtained from $T$ by inserting horizontal arcs. It then follows that $N_u$ is also an orchard since $N_d$ can be obtained from $N_u$ by inserting a root and orienting all edges. Alternatively, it can be seen that $N_d$ is an orchard since we can, for example, reduce reticulated cherries on $\langle c, b \rangle$ and $\langle e, d \rangle$to obtain the tree $T$, which can then be fully reduced by reducing cherries. Similarly, $N_u$ is an orchard since we can reduce 2-chains on $\langle b,c \rangle$ and $\langle d,e \rangle$ and subsequently reduce cherries in the resulting tree.
• Figure 2: (a) The root widget (the sheep). The widget consists of the vertices and edges in the shaded region. The dashed edges are connections to vertices in other widgets. (b) Its BPO. Arcs are smooth lines; edges are zigzag lines.
• Figure 3: (a) A wire widget. The widget consists of the vertices and edges in the shaded region. The dashed edges are connections to vertices in other widgets. (b) its True-BPO. (c) Its False-BPO. In figures (b) and (c), arcs are smooth lines; edges are zigzag lines. The orientations of the edges $\{o_1, o_2\}$ and $\{i_1, i_2\}$ are not shown in figures (b) and (c), as they are determined by the other widgets containing these edges.
• Figure 4: (a) A choice widget. The widget consists of the vertices and edges in the shaded region. The dashed edges represent wire widgets joining its inputs and outputs to other widgets. (b) Its True-BPO. (c) Its False-BPO. In figures (b) and (c), arcs are smooth lines; edges are zigzag lines.
• Figure 5: (a) A basic and-widget. The widget consists of the vertices and edges in the shaded region. The dashed edges represent wire widgets joining its inputs and outputs to other widgets. (b) Its BPO. Arcs are smooth lines; edges are zigzag lines.

Theorems & Definitions (36)

• Theorem 2.1
• proof
• Theorem 3.1: Van Iersel et al. van2022orchard
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.6