# Cluster Edge Deletion on 2-trees [closed]

Definitions: Cluster Edge Deletion problem is a graph modification problem, in which we want to remove the minimum number of edges such that the resulting graph does not contain a $P_3$ as an induced subgraph (that is, the resulting graph is a disjoin union of cliques).

The class of $k$-trees is defined as follows: a complete graph with $k$ vertices is a $k$-tree; a $k$-trees with $n + 1$ vertices $(n > k)$ can be constructed from a $k$-tree $T$ with $n$ vertices by adding a vertex adjacent to all vertices of a $k$-clique of $T$, and only to these vertices.

Question: Does exist a polynomial-time algorithm to solve Cluster Edge Deletion on 2-trees?

My idea: Let $G$ be the 2-tree. I transform $G$ to a new graph $G'$ where, each node of $G'$ represents a $K_3$ in $G$. And two nodes of $G'$ are adjacent if and only if their corresponding $K_3$ share a common vertex in $G$.

Let $M$ be the resulting graph after removing the minimum edges from $G$. I think that if always exist some graph $M$ contains a 3-clique $H$ which has a node with minimum degree in $G'$, then I can solve this problem on $G \backslash H$.

For example as below figure, I choose node $x$ with minimum degree 3 in $G'$, and after remove all vertices of clique $\{f,g,e\}$ in $G$, I got an union disjoint of two clique $\{f,g,e\}$ and $\{b,c,d\}$ in $G$. 