ISIPhyNC - Network #7

JSON code of the network

{"nodes":["r","a","b","c","d","e","f","1","2","3","4"],"edges": [["r","a"],["a","b"],["b","1"],["a","c"],["c","2"],["c","d"],["d","3"],["r","e"],["e","b"],["e","f"],["f","d"],["f","4"]]}

Visualization of the network

r is the root and the leaves are labeled from 1 to 4. Arcs are oriented from the parent to the child.

Classes containing this network or not

Classes which contain this network (with direct proof)

• binary normal: 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.
• binary time-consistent: Here is a time-consistent labeling t:
  • t(r)=0
  • t(a)=t(b)=t(e)=1
  • t(c)=t(d)=t(f)=t(1)=2
  • t(2)=t(3)=t(4)=3
• binary level-2: Easy to check.
• binary spread-2: Any set on any order on 4 vertices is composed of at most 2 intervals.
• binary galled network: The common arcs between the two reticulation cycles are (r,a) and (r,e), which are tree arcs.

Classes which do not contain this network (with direct proof)

• binary galled tree: Easy to check.
• binary nested: Easy to check.
• binary spread-1: Suppose by contradiction that there exists an order where there is exactly one interval for each cluster. Clusters {2,3} of vertex c and {3,4} of vertex f force 3 to be between 2 and 4 in this order. So 1 cannot be between 2 and 3 nor between 4 and 3. Therefore, either cluster {1,2,3} of vertex a and {1,3,4} of vertex e have to appear as 2 intervals in this order (split respectively by 4 or by 2).
• binary 2-reticulated: There are two reticulation vertices, b and d, which can be reached using two directed internally vertex disjoint paths from tree vertex r.

All classes

In the inclusion diagram below, the names of classes containing this network are colored green and the name of the classes not containing it are colored red.