Connectivity check in 3-connected planar graphs with obstacles

We define a vertex labelling for every planar 3-connected graph with n vertices from which one can answer connectivity queries. A connectivity query asks whether there exists in the given graph a path linking u and v that avoids a set F of edges and a set X of vertices. The vertices u,v and those of X are given by their labels. The edges of F are given by the labels of their ends. Each label has a size of O(log(n)) bits. Our construction makes an essential use of straight-line embeddings on n×n grids of simple loop-free planar graphs. Such embeddings can be constructed in linear time by Schnyder's algorithm [W. Schnyder. Embedding Planar Graphs in the Grid. SODA, First ACM-SIAM Symposium on Discrete Algorithms, San Francisco, pages 138–148, 1990] (see also [H. de Fraysseix, J. Pach, and R. Pollack. Small Sets Supporting Fary Embeddings of Planar Graphs. In Twentieth Annual ACM Symposium on Theory of Computing, pages 426–433. ACM, 1988; H. de Fraysseix, J. Pach, and R. Pollack. How to Draw a Planar Graph on a Grid. Combinatorica, 10:41–51, 1990]).