ISIPhyNC - Class: binary compressed

Definition

Relationships with other phylogenetic network classes

Maximum subclasses

• binary FU-stable
• binary reticulation-visible [reference] (See Proposition 1(2))

Minimum superclasses

• binary tree-based [reference] (The parents of any reticulation vertex of a binary compressed network N are not reticulation vertices, so according to Proposition 3, N is tree-based.)

Problems

Positive results proved for this class

No positive result found

Positive results deduced from superclasses

• Phylogenetic Network Isomorphism, positive result on binary: The phylogenetic network isomorphism problem can be solved in O(n⁴) 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

No negative result found

Negative results deduced from subclasses

• Cluster Containment, negative result on binary regular: 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, negative result on binary regular: NP-hard, reduction from Tree Containment on binary networks. [reference] (Theorem 3)

Properties

Properties proved for this class

No property found

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 #14 : None of the two reticulation vertices is parent of the other.

Deduced from class inclusions: network #5 (deduced from the inclusion of "binary unicyclic" in this class), network #14 (deduced from the inclusion of "binary galled network" in this class), network #6 (deduced from the inclusion of "binary galled tree" in this class), network #11 (deduced from the inclusion of "binary FU-stable" in this class), network #7 (deduced from the inclusion of "binary galled network" in this class), network #9 (deduced from the inclusion of "binary distinct-cluster" 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 #1 (deduced from the inclusion of "binary FU-stable" in this class), network #21 (deduced from the inclusion of "binary galled network" in this class), network #24 (deduced from the inclusion of "binary regular" in this class), network #2 (deduced from the inclusion of "binary galled network" in this class), network #8 (deduced from the inclusion of "binary nearly tree-child" in this class), network #4 (deduced from the inclusion of "binary FU-stable" in this class), network #18 (deduced from the inclusion of "binary galled network" in this class), network #15 (deduced from the inclusion of "binary normal" in this class)

Not in this class

Proved directly:
network #12 : Reticulation vertex a is a parent of reticulation vertex b.

Deduced from class inclusions: 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")