ISIPhyNC - Class: binary regular
Definition
A phylogenetic network is
binary regular if it is
binary and it is
decontracted-regular. [
reference]
Bibliographic references on the Who is who in phylogenetic networks
Relationships with other phylogenetic network classes
Maximum subclasses
Minimum superclasses
- binary distinct-cluster (A Hasse diagram of a family F of subsets of X does not contain two vertices corresponding to the same set of F, so a regular network does not contain two vertices with the same cluster.)
Problems
Positive results proved for this class
Positive results deduced from superclasses
- Phylogenetic Network Isomorphism, positive result on binary: The phylogenetic network isomorphism problem can be solved in O(n^{4}) time on binary networks. Simulations show that the algorithm is practical, with instances of 500 vertices solved in less than one tenth of a second. [reference]
Negative results proved for this class
- Cluster Containment: NP-hard, reduction from Cluster Containment in general graphs using:
- the procedures to refine the vertices of degree strictly greater than 3, described in the remark in the end of the proof of Theorem 4.1 of [KNTX2008] and in the proof of Observation 2 of [FIKS2015];
- the gadget in described in the proof of Theorem 3 of [ISS2010].
- Tree Containment: NP-hard, reduction from Tree Containment on binary networks. [reference] (Theorem 3)
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 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)
Properties deduced from superclasses
No property could be deduced from superclasses.
Properties deduced from subclasses
No property could be deduced from subclasses.
Examples of networks
In this class
proved directly:
network #24 : If we contract each reticulation vertex of this network with its child, we obtain the Hasse diagram of {{1},{2},{3},{4},{1,2},{2,3},{3,4},{1,2,3},{2,3,4},{1,2,3,4}}.
network #2 : If we contract each reticulation vertex of this network with its child, we obtain the Hasse diagram of {{1},{2},{3},{4},{5},{1,2},{2,3},{1,2,3},{3,4},{1,2,3,4},{4,5},{1,2,3,4,5}}.
network #8 : If we contract each reticulation vertex of this network with its child, we obtain the Hasse diagram of {{1},{2},{3},{4},{5},{1,2},{2,3},{3,4},{2,3,4},{3,4,5},{1,2,3,4},{1,2,3,4,5}}.
Deduced from class inclusions: network #7 (deduced from the inclusion of "binary normal" in this class), network #13 (deduced from the inclusion of "binary normal" in this class), network #22 (deduced from the inclusion of "binary normal" in this class), network #15 (deduced from the inclusion of "binary normal" in this class)
Not in this class
Proved directly:
network #9 : c(b)={2,3} is contained in c(e)={2,3,4}, however there is no path from e to b.
Deduced from class inclusions: network #5 (deduced from the inclusion of this class in "binary distinct-cluster"), 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 #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 #3 (deduced from the inclusion of this class in "binary tree-based"), network #4 (deduced from the inclusion of this class in "binary distinct-cluster")
About this website
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.
How to cite
P. Gambette, M. Morgado, N. Tavassoli & M. Weller (2018)
ISIPhyNC, an Information System on Inclusions of Phylogenetic Network Classes, manuscript in preparation.
Database content
73 classes of phylogenetic networks including 35 classes of binary phylogenetic networks (defined in a total of 20 bibliographic references), 51 inclusion relationships proved directly between classes (including some found in a total of 9 bibliographic references), 24 networks (68 memberships to a class, 56 non-memberships to a class), 3 problems considered, 3 properties considered, 37 theorems proved directly (including some found in a total of 17 bibliographic references) including 26 positive results (which can be extended to subclasses) and 11 negative results (which can be extended to superclasses).