Mathematics – Combinatorics
Scientific paper
2009-07-14
Mathematics
Combinatorics
27 pages
Scientific paper
The study of the structural properties of large random planar graphs has become in recent years a field of intense research in computer science and discrete mathematics. Nowadays, a random planar graph is an important and challenging model for evaluating methods that are developed to study properties of random graphs from classes with structural side constraints. In this paper we focus on the structure of random biconnected planar graphs regarding the sizes of their 3-connected building blocks, which we call cores. In fact, we prove a general theorem regarding random biconnected graphs. If B_n is a graph drawn uniformly at random from a class B of labeled biconnected graphs, then we show that with probability 1-o(1) B_n belongs to exactly one of the following categories: (i) Either there is a unique giant core in B_n, that is, there is a 0 < c < 1 such that the largest core contains ~ cn vertices, and every other core contains at most n^a vertices, where 0 < a < 1; (ii) or all cores of B_n contain O(log n) vertices. Moreover, we find the critical condition that determines the category to which B_n belongs, and also provide sharp concentration results for the counts of cores of all sizes between 1 and n. As a corollary, we obtain that a random biconnected planar graph belongs to category (i), where in particular c = 0.765... and a = 2/3.
Fountoulakis Nikolaos
Panagiotou Konstantinos
No associations
LandOfFree
3-Connected Cores In Random Planar Graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with 3-Connected Cores In Random Planar Graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and 3-Connected Cores In Random Planar Graphs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-331709