Exact Algorithms for Weighted Coloring in Special Classes of Tree and Cactus Graphs

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We study the weighted vertex coloring problem (WVCP) in binary trees and a restricted class of cactus graphs we called cactus paths. WVCP is a generalization of the vertex coloring problem where a color class of a feasible coloring is assigned a cost equal to the largest weight of a vertex from the color class. The objective is to find a feasible coloring which minimizes the sum of the color costs assigned to each color class. We improve the exact algorithms for solving WVCP on binary trees and propose new and efficient algorithms for WVCP on cycles and cactus paths with maximum degree three. Our work extends the results of Kavitha and Mestre [8]. Our algorithms have a time complexity of \(O(n \log ^2 n)\) for cactus paths and \(O(n^2 \log n)\) for binary trees.