Fig. 2.

An example reduction from the Clique problem to MCT. (a) An undirected graph H with $|E(H)|=4$ edges and $n=|V(H)|=4$ vertices, containing a clique of size 3. (b–e) The n = 4 input trees $\mathcal{T}=\{T_1,T_2,T_3,T_4\}$ to the MCT problem obtained from H. The problem instance of determining whether H contains a clique of size c = 3 reduces to the MCT instance $(\mathcal{T},k)$ where $k=n-c+1=2$. An optimal clustering σ for $(\mathcal{T},2)$ yields ${\mathcal{T}}_1=\{T_1,T_2,T_3\}$ and ${\mathcal{T}}_2=\left\{T_4\right\}$. (f) The parent-child graph $G_{{\mathcal{T}}_1}$, with the optimal consensus tree R₁ for input trees ${\mathcal{T}}_1$ indicated in red. The parent-child graph of ${\mathcal{T}}_2$ is identical to T₄ with edge weights $\ell (u,v)=1$ for each edge (u, v) such that the corresponding optimal consensus tree R₂ equals T₄. As such, the total distance equals $2[(c-1)·|E(H)|-\left(\begin{array}{c}c\\ 2\end{array}\right)]=10$. By Lemma 6, H contains a clique of size c = 3