ISIPhyNC - Class: binary galled tree

Definition

Relationships with other phylogenetic network classes

Maximum subclasses

• binary phylogenetic tree (A binary phylogenetic tree is a binary galled tree with no gall.)
• binary unicyclic (A binary unicyclic network is a binary galled tree with exactly one reticulation vertex.)

Minimum superclasses

• binary 2-nested
• binary galled network
• binary leaf outerplanar (It is possible to draw a binary galled tree with no leaf inside the galls in order to obtain an outerplanar realization.)
• binary level-2 (A binary galled tree is actually a binary level-1 network.)
• binary tree-child [reference] (A detailed proof is given in Lemma 6.11.11 of this book.)

Problems

Positive results proved for this class

No positive result found

Positive results deduced from superclasses

• Cluster Containment, positive result on binary level-2: Solvable in O(n) time [reference] (Observation 1)
• Cluster Containment, positive result on binary galled network: Solvable in polynomial time. [reference] (Lemma 1. The proof is a bit more detailed in Proposition 12 of this reference.)
• Cluster Containment, positive result on binary reticulation-visible: Solvable in O(n) time [reference]
• Cluster Containment, positive result on binary level-3: Solvable in O(n) time [reference] (Observation 1)
• Cluster Containment, positive result on binary level-k: Solvable in O(2^k.n) time [reference] (Observation 1)
• 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]
• Tree Containment, positive result on binary level-2: Solvable in O(n) time [reference] (Observation 1)
• Tree Containment, positive result on binary reticulation-visible: Solvable in O(n³) time [reference]
• Tree Containment, positive result on binary reticulation-visible: Solvable in O(n) time [reference]
• Tree Containment, positive result on binary reticulation-visible: Solvable in O(n) time [reference]
• Tree Containment, positive result on binary level-3: Solvable in O(n) time [reference] (Observation 1)
• Tree Containment, positive result on binary level-k: Solvable in O(2^k.n) time [reference] (Observation 1)
• Tree Containment, positive result on binary tree-child: Solvable in polynomial time [reference] (Theorem 1)
• Tree Containment, positive result on binary nearly stable: Solvable in O(n²) time [reference]
• Tree Containment, positive result on binary nearly stable: Solvable in O(n log n) time [reference]
• Tree Containment, positive result on binary nearly stable: Solvable in O(n) time [reference]

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

No property found

Properties deduced from superclasses

• Upper bound on the number of vertices, property of binary galled network: An upper bound on the number of vertices is 6n-5 and an upper bound on the number of reticulation vertices is 2n-2. Both bounds are tight. [reference] (Theorem 4.1, Figure 3 and Table 1)
• Upper bound on the number of vertices, property of binary reticulation-visible: An upper bound on the number of vertices is 8n-7 and an upper bound on the number of reticulation vertices is 3n-3. Both bounds are tight. [reference] (Theorem 1.2)
• Upper bound on the number of vertices, property of binary tree-child: An upper bound on the number of vertices is 5n-2. [reference] (Proof of Theorem 2)
• Upper bound on the number of vertices, property of binary nearly tree-child: An upper bound on the number of vertices is 4n. [reference] (Lemma 4)
• Upper bound on the number of vertices, property of binary nearly stable: An upper bound on the number of vertices is 26n-24. [reference] (Theorem 2 (adding the number of reticulation vertices, tree vertices, the root and the leaves))
• Upper bound on the number of vertices, property of binary nearly stable: An upper bound on the number of vertices is 8n-7 and an upper bound on the number of reticulation vertices is 3n-3. Both bounds are tight. [reference] (Theorem 5.5 and Table 1)
• Upper bound on the number of vertices, property of binary stable-child: An upper bound on the number of vertices is 16n-15 and an upper bound on the number of reticulation vertices is 7n-7. Both bounds are tight. [reference] (Theorem 6.4 and Table 1)

Properties deduced from subclasses

No property could be deduced from subclasses.

Examples of networks

In this class

proved directly:
network #6 :

Deduced from class inclusions: network #5 (deduced from the inclusion of "binary unicyclic" in this class)

Not in this class

Proved directly:
network #7 :

Deduced from class inclusions: network #20 (deduced from the inclusion of this class in "binary 3-nested"), network #12 (deduced from the inclusion of this class in "binary compressed"), network #17 (deduced from the inclusion of this class in "binary nearly stable"), network #14 (deduced from the inclusion of this class in "binary tree-sibling"), network #11 (deduced from the inclusion of this class in "binary level-2"), network #9 (deduced from the inclusion of this class in "binary leaf outerplanar"), network #13 (deduced from the inclusion of this class in "binary galled network"), network #7 (deduced from the inclusion of this class in "binary spread-1"), network #1 (deduced from the inclusion of this class in "binary level-2"), network #10 (deduced from the inclusion of this class in "binary tree-based"), network #21 (deduced from the inclusion of this class in "binary nearly stable"), network #24 (deduced from the inclusion of this class in "binary reticulation-visible"), network #19 (deduced from the inclusion of this class in "binary 2-nested"), network #3 (deduced from the inclusion of this class in "binary level-2"), network #2 (deduced from the inclusion of this class in "binary level-2"), network #8 (deduced from the inclusion of this class in "binary level-2"), network #4 (deduced from the inclusion of this class in "binary galled network"), network #16 (deduced from the inclusion of this class in "binary level-3"), network #18 (deduced from the inclusion of this class in "binary level-3"), network #23 (deduced from the inclusion of this class in "binary level-3"), network #15 (deduced from the inclusion of this class in "binary level-3")