Merging Graph Edges to form a 2 color-able Graph (with weight constraints)? Given an undirected graph $G$. Each vertex has a weight 1.

We define shrinking an edge as merging and replacing 2 adjacent vertices $(A, B)$ with a new vertex $C$ such that all the vertices that were connected to $A$ or $B$ are now connected to $C$. Moreover the weight of the new vertex $C$ is sum of weight of $A$ and $B$.

Question: Determine if for a given graph $G$, we can merge vertices such that in the new Graph:


Each vertex has a weight 2.
The new graph is 2 color-able.


How do we approach this problem?
Your problem is exactly equivalent to the 2-colorable perfect matching problem:
On input a graph, color the vertices of the graph with two colors such that each vertex has exactly one neighbor of the same color.

The 2-colorable perfect matching problem was introduced in "The complexity of satisfiability problems" by T. J. Schaefer and is NP-complete.