I am exploring a reduction from the general Independent Set Problem to the Independent Set Problem specifically for 3-colorable graphs. The goal is to demonstrate that the maximal independent set of a general graph can be determined using an algorithm designed for 3-colorable graphs.

Construction Approach: Given a graph G=(V,E), I construct a new graph G′ by modifying each edge e∈E. For an edge e=(u,v), I introduce two new vertices we and xe. The edge (u,v) is then replaced with a path u−w−x−v.

Coloring Argument: Graph G′ can be shown to be 3-colorable. By coloring the original vertices from G with color 1, and alternating the colors 2 and 3 for the newly added vertices on each edge, a valid 3-coloring for G′ is achieved.

Question: Assuming we have determined the size of the maximal independent set in the 3-colorable graph G′, how can we utilize this information to find the size of the maximal independent set in the original graph G?

I find myself at an impasse with this question. The strategy for constructing G’ was suggested by my instructor, and I have followed it to the best of my understanding. However, I am still seeking clarity on how to proceed further. Any guidance or insights provided would be immensely appreciated!