A dynamic connectivity problem consists of an initial graph, and a sequence of operations consisting of graph modifications and graph connectivity tests. The size n of the problem is the sum of the maximum number of v...
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A dynamic connectivity problem consists of an initial graph, and a sequence of operations consisting of graph modifications and graph connectivity tests. The size n of the problem is the sum of the maximum number of vertices and edges of the derived graph, plus the number of operations to be executed. Each graph modification is a deletion of either an edge or an isolated vertex. Each graph connectivity test is to determine if there exists a path in the current graph between two given vertices (the vertices can vary for distinct tests). The best previously known time for this dynamic connectivity problem was Ω(n 2 ). Our main result is an O(ng+n log n) time algorithm for the dynamic connectivity problem in the case of the maximum genus of the derived graph being g.
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