The vertex coloring problem (VCP) is a classical problem of assigning a color to each vertex of a graph such that no two adjacent vertices have the same color. In the weighted version of classical VCP, we are given an undirected weighted graph $G = (V,E)$ where each vertex $v \in V$ is assigned a positive weight $w_v$. The objective of the

weighted vertex coloring problem (WVCP) is to minimize the sum of the costs of the colors from a feasible vertex coloring. The cost of a color equals the maximum vertex weight among all vertices assigned to the color.

We study the WVCP in binary trees and a restricted class of cactus graphs we called cactus paths. 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. We introduce the concept of feasible weight sets which proves to be

useful in solving WVCP exactly on a wider range of graphs. Our algorithms have a time complexity of $O(n \log^2 n)$ for cactus paths and $O(n^2 \log n)$ for binary trees. (This is a joint work with Robert Benkoczi and Daya Gaur).