On the existence of funneled orientations for classes of rooted phylogenetic networks

TL;DR

This work resolves open questions about orienting unrooted phylogenetic networks into funnelled rooted networks by proving NP-completeness for multiple rooted classes, including tree-child, tree-sibling, reticulation-visible, and normal, even when internal degrees are bounded by $5$ and without a fixed root. The authors correct a flawed proof from prior work and deliver a robust gadget-based reduction framework leveraging Positive Not-All-Equal $(2,3)$-SAT and Positive 1-in-3 SAT, with three key gadgets guiding the reductions. The results hold under both rooting variants (A and B) and extend from pseudo to true phylogenetic networks, via degree-2 suppression techniques, clarifying the computational boundaries for funnelled orientations. Overall, the paper sharpens the complexity landscape for orienting unrooted networks and informs algorithmic limits in phylogenetic inference.

Abstract

Recently, there has been a growing interest in the relationships between unrooted and rooted phylogenetic networks. In this context, a natural question to ask is if an unrooted phylogenetic network U can be oriented as a rooted phylogenetic network such that the latter satisfies certain structural properties. In a recent preprint, Bulteau et al. claim that it is computational hard to decide if U has a funneled (resp. funneled tree-child) orientation, for when the internal vertices of U have degree at most 5. Unfortunately, the proof of their funneled tree-child result appears to be incorrect. In this paper, we present a corrected proof and show that hardness remains for other popular classes of rooted phylogenetic networks such as funneled normal and funneled reticulation-visible. Additionally, our results hold regardless of whether U is rooted at an existing vertex or by subdividing an edge with the root.

Figures (10)

• Figure 1: (i) Two connector networks ${\mathcal{G}}_1$ and ${\mathcal{G}}'_1$ each with a single connector leaf $r$. (ii) A rooted pseudo network ${\mathcal{N}}$ obtained by identifying $r$ in ${\mathcal{G}}_1$ with $r$ in ${\mathcal{G}}'_1$ and orienting the resulting unrooted pseudo network according to Variant $A$. Observe that ${\mathcal{N}}$ is not tree-child. (iii) An orientation of ${\mathcal{G}}_1$ following Variant $A$. (iv) An orientation of ${\mathcal{G}}_1$ following Variant $B$. As we will see in Lemma \ref{['lem:root_gadget']}, for ${\mathcal{C}}$ being the class of tree-child network, ${\mathcal{G}}_1$ is ${\mathcal{C}}_A$-root-forcing because every ${\mathcal{C}}_A$-orientation of ${\mathcal{G}}_1$ places the root on an edge that is not $\{r,v\}$.
• Figure 2: (a) The construction of an unrooted pseudo network ${\mathcal{U}}$ for the Positive Not-All-Equal $(2,3)$-SAT instance (1), and (b) an orientation ${\mathcal{O}}$ of ${\mathcal{U}}$ that is rooted at $r$. Observe that ${\mathcal{O}}$ is funneled but not tree-child.
• Figure 3: Left: Root gadget with a single connector leaf $r$, and two non-connector leaves $\ell$ and $\ell'$. Right: An orientation of the root gadget.
• Figure 4: Connection gadget (a) with two connector leaves $s$ and $t$, and two non-connector leaves $\ell$ and $\ell'$. Two orientations (b) and (c) of the connection gadget. Both orientations are ${\mathcal{F}}$-compatible and strongly $\mathcal{TC}$-compatible.
• Figure 5: Left: Caterpillar gadget with $n+1$ connector leaves, and $1+2n$ non-connector leaves. Right: An ${\mathcal{F}}$-compatible and strongly $\mathcal{TC}$-compatible orientation of the caterpillar gadget.

Theorems & Definitions (26)

• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.5
• proof
• Theorem 3.6
• proof