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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 tosolve the same problem for several classes of graphs, defined by forbiddeninduced subgraphs, such as (diamond, co-diamond)-free graphs.

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