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Constructing rooted phylogenetic networks from rooted phylogenetic trees has become an important problem in molecular evolution. So far, many methods have been presented in this area, in which most efficient methods are based on the incompatible graph, such as the C

The evolutionary history of species is usually represented as a (rooted) phylogenetic tree, in which one species has only one parent. Actually, the evolution of species has caused reticulate events such as hybridizations, horizontal gene transfers, and recombinations [

Phylogenetic networks can be classified into unrooted [

The algorithms constructing rooted phylogenetic networks from rooted phylogenetic trees are mainly classified into three types: the cluster network [

Let

Two rooted phylogenetic trees

The abovementioned three types of methods constructing networks are based on clusters; that is, they first compute all of the clusters represented by input trees and then construct a network representing all clusters in the softwired sense. In this process, the third type of methods (C

A rooted phylogenetic network

Given a set of taxa

Given a cluster set

For each maximal ST-set

Let

for any two clusters

Then, we also say that the network

Given a set of clusters

A network constructed by the DC algorithms for the set of clusters in Figure

The C

All networks constructed by the C

Two networks

the label of

Given two sets of clusters

By Definition

Given a DC network

From the constructing process of DC networks, this conclusion is obvious.

Let

There must exist a DC network

Then, we can obtain a network

For two isomorphic sets of clusters

Let

Any one element

Each one of

For a cluster set

Let

The topology of the linear biconnected component with three nodes.

Figure

Any one set of clusters in

Figure

The DC networks for all simplest cluster sets whose incompatible graphs are topologies in Figure

Let

The topology of the nonlinear biconnected component with three nodes.

Figure

If

Figure

The DC networks for all simplest cluster sets whose incompatible graphs are topologies in Figure

This paper computes all simplest sets of clusters for the topologies of incompatible graph with two nodes and three nodes. We can construct the DC networks for those simplest sets of clusters and save them. When constructing DC networks for any one set of clusters

We will compute the simplest sets of clusters for more topologies of incompatible graph in the future.

The authors declare that they have no competing interests.

The work was supported by the Natural Science Foundation of Inner Mongolia Province of China (2015BS0601) and the National Natural Science Foundation of China (61300098, 31360289).