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A Category Sequence Traversal Query (CSTQ) is a new type of category based query that determines the minimum-cost path with a predefined origin-destination pair, traversing at least a single selection from each of a set of categories in a specified order. CSTQ has distinct applications in transportation trip planning and other potential applications whenever a workflow exists to traverse a set of categories of interest. This paper proposes bivariate best first searches to process CSTQ in a transportation network. Specifically, this paper 1) proposes a general multivariate best first search framework, L#, to process multiple points/categories of interest in a graph; 2) presents a developed bivariate best first search algorithm, C*, to process CSTQ in a general graph; and 3) provides C*-P and C*-Dijkstra, two bivariate best-first-search instances of C*, to retrieve optimal solutions for CSTQ processing in a transportation network. The performance of C*-P and C*-Dijkstra are analyzed with a set of experiments in a large urban dense transportation network, and the results show that on average, C*-Dijkstra is 5.9~8.8 times slower and expands 4.3~5.0 times more states than C*-P, with number of categories up to 7 and number of objects in each category up to 3800.