Maximal Biconnected Subgraphs of Random Planar Graphs

Let C be a class of labeled connected graphs, and let C-n be a graph drawn uniformly at random from graphs in C that contain exactly n vertices. Denote by b(l; C-n) the number of blocks (i.e., maximal biconnected subgraphs) of C-n that contain exactly l vertices, and let lb(C-n) be the number of vertices in a largest block of C-n. We show that under certain general assumptions on C, C-n belongs with high probability to one of the following categories: (1) lb(C-n) similar to cn, for some explicitly given c = c( C), and the second largest block is of order n(alpha), where 1 > alpha = alpha(C), or (2) lb(C-n) = O(log n), that is, all blocks contain at most logarithmically many vertices. Moreover, in both cases we show that the quantity b(l; C-n) is concentrated for all l, and we determine its expected value. As a corollary we obtain that the class of planar graphs belongs to category (1). In contrast to that, outerplanar and series-parallel graphs belong to category (2).