A characterization of terminal planar networks by forbidden structures

Abstract

The class of terminal planar networks was recently introduced from a biological perspective in relation to the visualization of phylogenetic networks, and its connection to upward planar networks has been established. We provide a Kuratowski-type theorem that characterizes terminal planar networks by a finite set of forbidden structures, defined via six families of 0/1-labeled graphs. Another characterization based on planarity of supergraphs yields linear-time algorithms for testing terminal planarity and for computing such planar drawings. We describe an application that is potentially relevant in broader, non-phylogenetic settings. We also discuss a connection of our main result to an open problem on the forbidden structures of single-source upward planar networks.

Figures (12)

• Figure 1: An illustration of Theorem \ref{['thm:up_pst']}. Left: an upward planar digraph $G$. Right: an acyclic planar $st$-digraph that contains $G$ as a spanning subgraph.
• Figure 2: The undirected phylogenetic networks $G_1$ and $G_2$ mentioned in Remark \ref{['rem:no.null']}: $G_2$ can be oriented into a directed phylogenetic network while $G_1$ cannot.
• Figure 3: A directed phylogenetic network $N$ with source $s$ and sink set $\{t_1, t_2, t_3, t_4\}$ (left) and the $t$-completion $N ^+$ of $N$ (right).
• Figure 4: Examples demonstrating the strict inclusions expressed in \ref{['eq:inclusion']}. $N_1\in \mathcal{P}_{\mathrm{outer}}(S)$, $N_2\in \mathcal{P}_{\mathrm{terminal}}(S)\setminus \mathcal{P}_{\mathrm{outer}}(S)$, $N_3\in \mathcal{P}_{\mathrm{upward}}(S)\setminus \mathcal{P}_{\mathrm{terminal}}(S)$, $N_4\in \mathcal{P}(S)\setminus \mathcal{P}_{\mathrm{upward}}(S)$, $N_5 \not \in \mathcal{P}(S)$ (all examples except $N_1$ are reproduced from Fig. 2 of WuMoulton9804871).
• Figure 5: Illustration of cut-labeled graphs and Definition \ref{['dfn:cut_completion']}. i) A directed phylogenetic network $N$ with source $s$ and sink set $\{t_1, t_2, t_3,t_4\}$. ii) The cut-completion $N^c$ of $N$. iii) The cut-labeled graph $L(N_u)$ of $N$. iv) The label-preserving cut-completion $L^c(N_u)$ of $L(N_u)$.

Theorems & Definitions (40)

• Theorem 3.1: Kuratowski1930
• Theorem 3.2: Wagner1937
• Theorem 3.3: AIHPB_1967__3_4_433_0
• Definition 3.4
• Theorem 3.5: DIBATTISTA1988175Kelly1987; see also Theorem 1 in garg1995upward
• Definition 3.6
• Remark 3.7
• Definition 3.8
• Definition 3.9: WuMoulton9804871
• Theorem 3.10: WuMoulton9804871