# ISIPhyNC - Class: binary normal

## Definition

A phylogenetic network is binary normal if it is binary and it is normal. [reference]

Bibliographic references on the Who is who in phylogenetic networks

## Relationships with other phylogenetic network classes

### Maximum subclasses

• binary phylogenetic tree (There is no redundant arc in a phylogenetic binary tree and every vertex has no reticulation vertices as children.)

## Problems

### Negative results proved for this class

No negative result found

### Negative results deduced from subclasses

No negative result could be deduced from subclasses.

## Properties

### Properties proved for this class

• Upper bound on the number of vertices: An upper bound on the number of vertices is n2-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)

### Properties deduced from subclasses

No property could be deduced from subclasses.

## Examples of networks

### In this class

proved directly:
network #7 : The network contains no redundant arc, and all parents of reticulations have a tree child: a and e, parents of b, have respectively c and f as tree child; e and f, parents of d, have respectively 2 and 4 as tree child.
network #13 : The network contains no redundant arc, and all parents of reticulations have a tree child: a and f, parents of d, have respectively b and 4 as tree child; b and e, parents of c, have respectively 1 and 3 as tree child.
network #22 : The network contains no redundant arc, and all parents of reticulations have a tree child: a and f, parents of b, have respectively c and 2 as tree child; c and e, parents of d, have respectively 3 and f as tree child.
network #15 : The network contains no redundant arc, and all parents of reticulations have a tree child: b and g, parents of c, have respectively 1 and h as tree child; d and j, parents of e, have respectively 2 and 4 as tree child; h and l, parents of i, have respectively 5 and 6 as tree child; f and m, parents of k, have respectively g and 7 as tree child; .

Deduced from class inclusions: no network found in this class using class inclusions

### Not in this class

Proved directly:
no network found outside this class with a direct proof

Deduced from class inclusions: network #5 (deduced from the inclusion of this class in "binary distinct-cluster"), network #17 (deduced from the inclusion of this class in "binary nearly stable"), network #14 (deduced from the inclusion of this class in "binary FU-stable"), network #12 (deduced from the inclusion of this class in "binary compressed"), network #6 (deduced from the inclusion of this class in "binary distinct-cluster"), network #11 (deduced from the inclusion of this class in "binary distinct-cluster"), network #9 (deduced from the inclusion of this class in "binary regular"), network #21 (deduced from the inclusion of this class in "binary nearly stable"), network #1 (deduced from the inclusion of this class in "binary distinct-cluster"), network #10 (deduced from the inclusion of this class in "binary tree-based"), network #24 (deduced from the inclusion of this class in "binary reticulation-visible"), network #19 (deduced from the inclusion of this class in "binary nearly stable"), network #8 (deduced from the inclusion of this class in "binary nearly stable"), network #2 (deduced from the inclusion of this class in "binary tree-sibling"), network #3 (deduced from the inclusion of this class in "binary tree-based"), network #4 (deduced from the inclusion of this class in "binary nearly tree-child")

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.