15 problems

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…

Limit law conjecture. The following limit behavior holds:

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-…

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…

Let $X^{(\ell)}_k$ denote the number of lineages in the ancestry of a leaf after moving $k$ steps away from that leaf in a uniformly sampled ranked tree-child network with $\ell$ l…

Let $N(B)$ be a tree-metrizable minimal support network of a biconnected component $B$, and suppose that $N(B)$ has $n$ leaves. A reticulation arc is an arc entering a reticulation…

Let $N$ be a level-$k$ HGT network with a biconnected component $B$. Let $N(B)$ denote the minimal support network of $B$, let $T_{N(B)}$ denote its underlying tree, and call a pai…

Let $N$ be a rooted phylogenetic network on leaf set $X$. The quantities $\ell(N)$ and $e(N)$ measure, respectively, the tree-based proximity of $N$ and the minimum number of direc…

Let $N$ and $N'$ be non-binary tree-child networks on the same leaf set. A tree-child sequence (TCS) of $N$ is a sequence of reductions associated with the tree-child network. Univ…

Let $N$ and $N'$ be CPNs on the same leaf-set in the $(, )$ class. Here, a cherry-picking sequence (CPS) of $N$ is a sequence used to reduce $N'$, and $N'$ is contained in $N$ acco…

MLLS-reconstructibility conjecture. The class of binary valid networks is MLLS-reconstructible.