Planar Graph Decomposition and All Pairs Shortest Paths

An algorithm is presented for generating a succinct encoding of all pares shortest path reformation in a directed planar graph G with real-valued edge costs but no negative cycles. The algorithm runs m O(prI ) time, where n M the number of vertices in G, and p is the mimmum cardinalit y of a subset of the faces that cover all vertices, taken over all planar embeddings of G. The algorithm is based on a decomposition of the graph into O(pn ) outerplanar subgraphs satisfying certain separator properties. Lmear-time algorithms are presented for various subproblems including that of finding an appropriate embedding of G and a corresponding face-on-vertex covering of cardinabty 0(p). and of generating all pairs shortest path Information in a directed outerplanar graph.

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