proved directly:
network #9 : Consider the order 1 2 3 4 of the leaves to obtain spread 1.
Deduced from class inclusions: network #5 (deduced from the inclusion of "binary unicyclic" in this class), network #12 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #6 (deduced from the inclusion of "binary galled tree" in this class), network #11 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #13 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #1 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #10 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #2 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #8 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #4 (deduced from the inclusion of "binary leaf outerplanar" in this class), network #16 (deduced from the inclusion of "binary leaf outerplanar" in this class)
Proved directly:
network #7 : 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).
network #3 : Suppose there exists an order of the leaves which corresponds to spread 1. Vertices b, e, g, i are all above leaf 2 as well as above another leaf which is not below the other ones. So whatever two leaves, say 1 and 3, we choose to be around leaf 2 in the order, 4 and 5 will not be, so the leaves below g and h will not be an interval in this order.
Deduced from class inclusions: no network found outside this class using class inclusions
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.