Reducing the Independent Set Problem to Independent Set for 3-Colorable Graphs

Question

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 \in E $. For an edge $ e=(u, v) $, I introduce two new vertices $ w_e $ and $ x_e $. The edge $ (u, v) $ is then replaced with a path $ u - w_e - x_e - 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!

Answer 1

The independent set problem for 3-colorable graphs has been discussed here. More specifically, the link discusses the hardness of the maximum independent set problem and the 3-colorability problem.

Suppose we find a maximal independent set for $G'$ and let it be $I'$. We will now prune $I'$ so that it becomes an independent set of $G$. For each edge $(u,v)$ in the origianl graph $G$, we may have exactly two vertices among $\{u,v,w_e,x_e\}$ in $I'$. More specifically, we can have either $\{u,x_e\} \in I'$ or $\{v,w_e\} \in I'$ or $\{u,v\} \in I'$. For the first two cases, we can simply delete either $w_e$ or $x_e$. For the case $\{u,v\} \in I'$, we have to delete either $u$ or $v$ in order to make it an independent set of $G$. Here, we can apply any greedy criterion, such as deleting the one with highest degree in $G$. At the end of this iterative deletion, we will be left with a maximal independent set of $G$.