Polynomial-time algorithms for minimum weighted colorings of ($P_5, \bar{P}_5$)-free graphs and related graph classes

TL;DR

This paper presents an $O(n^3)$ algorithm for minimum weighted coloring of ($P_5, ar{P}_5$)-free graphs and extends the technique to other graph classes defined by forbidden subgraphs.

Contribution

It introduces a polynomial-time algorithm for coloring specific graph classes and generalizes the method to related classes with forbidden induced subgraphs.

Findings

Efficient $O(n^3)$ coloring algorithm for ($P_5, ar{P}_5$)-free graphs.

Extension of the algorithm to (diamond, co-diamond)-free graphs.

Demonstrates polynomial-time solutions for coloring problems in new graph classes.

Abstract

We design an $O(n^3)$ algorithm to find a minimum weighted coloring of a ($P_5, \bar{P}_5$)-free graph. Furthermore, the same technique can be used to solve the same problem for several classes of graphs, defined by forbidden induced subgraphs, such as (diamond, co-diamond)-free graphs.