Simultaneous separation in bounded degree trees

Abstract

It follows from a classical result of Jordan that every tree with maximum degree at most $r$ containing a vertex set labeled by $[n]$, has a single-edge cut which separates two subsets $A,B \subset [n]$ for which $\min\{|A|,|B|\} \ge (n-1)/r$. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of {\em several} trees: Given $k$ trees with maximum degree at most $r$, containing a common vertex set labeled by $[n]$, we ask for a single-edge cut in each tree which maximizes $min\{|A|,|B|\}$ where $A,B \subset [n]$ are separated by the corresponding cut at each tree. Denoting this maximum by $f(r,k,n)$ and considering the limit $f(r,k) = \lim_{n \rightarrow \infty} f(r,k,n)/n$ (which is shown to always exist) we determine that $f(r,2)=\frac{1}{2r}$ and determine that $f(3,3)=\frac{2}{27}$, which is already quite intricate. The case $r=3$ is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over $n$ taxa have a split at each tree which separates two taxa sets of order at least $n/6$ (resp. $2n/27$), and these bounds are asymptotically tight.

Figures (4)

• Figure 1: Two trees $T_1$ (left) and $T_2$ (right) with a common subset of vertices labeled by $[12]$ (in this case all vertices are common), both with maximum degree $3$. Any two disjoint subsets of vertices of size $3$ are not separated by a cut vector. This shows that $f(3,2,12) \le 2$. The depicted cut vector $(e_1,e_2)$ separates $X=\{1,3\}$ and $Y=\{5,6,7,9,10,11\}$. This shows that $f(\{T_1,T_2\},12) = 2$.
• Figure 2: The trees $T_1$ (left) and $T_2$ (right), the cut vector $(e_1,e_2)$, and the partition sets; recall that $X_{1,1} = \{y\} \cup X_{1,1,0} \cup X_{1,1,1}$.
• Figure 3: An unrooted phylogenetic tree $H$ on the left with $\ell=4$ leaves. The leaf $v$ is replaced with a $5$-leaf caterpillar (circled) to obtain the tree in the middle; here $x_v=4$. The tree on the right is a balanced caterpillar blowup of $H$.
• Figure 4: Three unrooted phylogenetic trees, each on taxa set $[10]$ with no common quartet (i.e., no disjoint taxa sets $X,Y$ of size $2$ each that are separated in all three trees).

Theorems & Definitions (14)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• Proposition 2.4
• proof