Bounding the Number of Reduced Trees, Cographs, and Series-Parallel Graphs by Compression

Abstract

We give an efficient encoding and decoding scheme for computing a compact representation of a graph in one of unordered reduced trees, cographs, and series-parallel graphs. The unordered reduced trees are rooted trees in which (i) the ordering of children of each vertex does not matter, and (ii) no vertex has exactly one child. This is one of basic models frequently used in many areas. Our algorithm computes a bit string of length 2l−1 for a given unordered reduced tree with l≥1 leaves in O(l) time, whereas a known folklore algorithm computes a bit string of length 2n−2 for an ordered tree with n vertices. Note that in an unordered reduced tree l≤n $\left\lceil2.5285m-2\right\rceil $ bits. Hence the number of series-parallel graphs with m edges is at most $2^{\left\lceil2.5285m-2\right\rceil }$ . As far as the authors know, this is the first non-trivial result about the number of series-parallel graphs. The encoding and decoding of the cographs and series-parallel graphs also can be done in linear time.