Merging Graph Edges to form a 2 color-able Graph (with weight constraints)?
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.