. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)-time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different trade-offs have been proposed. The adjacency-list encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted n-node tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...

Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.