45 problems

Let $N$ be a network. Tree-child proximity conjecture. … The conjecture asks whether the analogous inequality to the known result $A_{\mathcal{TC}}(N)\le L_{\mathcal{TC}}(N)$ holds…

Let $\mathcal{TC}_{n,k}$ be the class of tree-child networks with $n$ leaves and $k$ reticulation nodes, and let $\mathcal{C}_{n-1,k}$ be the class of words over the alphabet…

Let $\mathcal{TC}_{n,k}$ be the set of all tree-child networks with $n$ leaves and $k$ reticulation nodes. The numbers $a_{n-1,k}$ are the coefficients appearing in the paper's pre…

Let $X$ be the leaf set, and let $N_1$ and $N_2$ be binary normal networks on $X$ with no near-sibling reticulations. Write $R(N)$ for the set of rooted triples displayed by a netw…

Let $N$ be an outer-labeled planar, galled, level-$3$ bloblet network, and let its NANUQ metric be the pairwise distance metric obtained from the NANUQ construction. Level-3 NANUQ…

Let $N$ be an outer-labeled planar galled network of level $2$, and let its NANUQ metric be the pairwise distance metric obtained from the NANUQ construction. The theorem asserting…

Level-minimization NP-hardness conjecture. The level minimization problem for support networks is NP-hard.

Let $TC_{n,k}$ denote the class of binary tree-child networks with $k$ reticulations and $n$ leaves, and let $|TC_{n,k}|$ be its cardinality. Tree-child network recursion conjectur…

Let $k$ be a positive integer, and consider sets of three trees on $2^k$ taxa: two caterpillar trees and a fully balanced tree whose leaves are equally distant from the root. Maxim…

A semi-directed binary simple level-3 network is a semi-directed binary phylogenetic network of level $3$ with no additional simplicity violations, and let $N_1$ and $N_2$ denote t…

Orientation characterization conjecture. An undirected network is an orchard if and only if it can be oriented as a directed network that is an orchard.

Let $\mu_E$ be the edge-based representation of a semidirected network. Extend $\mu_E$ by adding, in the same way as the extended node-based representation, a coordinate that count…

A semidirected network is a network with directed and undirected edges as in the paper, and let $\mu_E$ denote its edge-based representation. Let $d_{\mu_E}$ be the associated diss…

An undirected phylogenetic network $N$ is given, and the Tree-child Orientation problem asks whether $N$ can be oriented to a tree-child network $\hat{N}$. NP-completeness conjectu…

A semi-directed phylogenetic network is a network with a fixed leaf set in which the five move types employed in SNaQ act on the network. SNaQ move-set conjecture. The five types o…

Let $N$ be a metric semidirected network on taxon set $X$, let $d_N$ denote its average distances on $X$, and let $T$ be the distance split tree reconstructed from $d_N$. Distance-…

A phylogenetic network has a tree of blobs, the tree obtained by contracting each biconnected component (blob) to a vertex. Tree-of-blobs identifiability conjecture. The tree of bl…

Let $\mathcal{TC}_{n,k}$ be the set of tree-child networks with $n$ leaves and $k$ reticulation nodes. Let $\mathcal{C}_{n-1,k}$ be the class of words on $n-1$ letters in which $k$…

Let $J_n$ be the ideal of invariants of the CFN model on the $n$-sunlet network, and let $I_n$ be the ideal generated by all quadratic invariants in $J_n$. Dimension and primality…

Let a level-$2$ network be a network whose blobs have level at most $2$, let a macaron be a level-$2$ blob with two cut-edges, and let an alt-path structure be the structure descri…

Let $\mathrm{GN}_{n,k}$ denote the number of galled networks with parameters $n$ and $k$. Monotonicity conjecture. The sequence $\mathrm{GN}_{n,k}$ is increasing for…

Exterior-shape invariance conjecture. If

2-nested network conjecture. The resistance distance of every 2-nested phylogenetic network is Kalmanson.

Realizability conjecture. If $d53d$ is faithfully phylogenetic, then there exists a weighted phylogenetic network $N$ such that

Outer planarity conjecture. Outer planarity is sufficient for the resistance distance to be Kalmanson.