The number of vertices is bounded by the number of leaves.

There exists a function *f* such that any network with *n* leaves has at most *f*(*n*) vertices.

- binary galled network: An upper bound on the number of vertices is 6
*n*-5 and an upper bound on the number of reticulation vertices is 2*n*-2. Both bounds are tight. [reference] (Theorem 4.1, Figure 3 and Table 1) - binary nearly stable: An upper bound on the number of vertices is 26
*n*-24. [reference] (Theorem 2 (adding the number of reticulation vertices, tree vertices, the root and the leaves)) - binary nearly stable: An upper bound on the number of vertices is 8
*n*-7 and an upper bound on the number of reticulation vertices is 3*n*-3. Both bounds are tight. [reference] (Theorem 5.5 and Table 1) - binary nearly tree-child: An upper bound on the number of vertices is 4
*n*. [reference] (Lemma 4) - binary normal: An upper bound on the number of vertices is
*n*^{2}-*n*+2 [reference] (Theorem 5.1(2), with a multiplication by 2 to take into account the number of vertices possibly added during the "decontraction" to obtain a binary phylogenetic network) - binary regular: An upper bound on the number of vertices is 2
^{n}. [reference] (Theorem 5.1(3), with a multiplication by 2 to take into account the number of vertices possibly added during the "decontraction" to obtain a binary phylogenetic network) - binary reticulation-visible: An upper bound on the number of vertices is 8
*n*-7 and an upper bound on the number of reticulation vertices is 3*n*-3. Both bounds are tight. [reference] (Theorem 1.2) - binary stable-child: An upper bound on the number of vertices is 16
*n*-15 and an upper bound on the number of reticulation vertices is 7*n*-7. Both bounds are tight. [reference] (Theorem 6.4 and Table 1) - binary tree-child: An upper bound on the number of vertices is 5
*n*-2. [reference] (Proof of Theorem 2)

In the inclusion diagram below, the names of classes with this property are colored.

This website was programmed and is maintained by Philippe Gambette. It was started during the internship of Maxime Morgado at LIGM, in June-July 2015, and also contains contributions made from Narges Tavassoli from November 2016 to January 2017.

Please contact Philippe Gambette if you have any suggestions about this website, especially about problems, properties, results or subclasses to add.