Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees

Ambivalent data structures are presented for several problems on undirected graphs. They are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log β(m,n)+min{k^3/2, km ^1/2}) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n+k(log n)³ ) time. Ambivalent data structures are also used to maintain dynamically 2-edge-connectivity information. Edges and vertices can be inserted or deleted in O(m^1/2) time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Again, the techniques are extended to maintain an embedded planar graph so that edges and vertices can be inserted or deleted in O((log n)³) time, and a query answered in O(log n) time