Some Properties of Binary Series-Parallel Graphs

Abstract

We introduce an incremental growth model for directed binary series-parallel (SP) graphs. The vertices of a directed binary SP graphs can only have outdgrees 1 or 2. We show that the number of vertices of outdegree 1 have a normal distribution (so, necessarily, the vertices of outdegree 2 have a normal distribution, too). Furthermore, we study the average length of a random walk between the poles of the graph. The asymptotic equivalent of the latter property includes the golden ratio. Pólya urns will systematically provide a coding method to initiate the studies.

Combinatorial Analysis of Growth Models for Series-Parallel Networks

“…We collect our findings for the r.v. L n in the next theorem, where we also restate the result for the expectation E(L n ) obtained in [13].…”

“…Extracting coefficients from it and applying Stirling's formula immediately yields the explicit and asymptotic result for the expectation obtained by Mahmoud in [13] and that is restated in the theorem.…”

“…We focus here on the quantities degree D n of the source and/or sink, length L n of a random sourceto-sink path and the number P n of source-to-sink paths in a random series-parallel network of size n, but mention that also other quantities (as, e.g., the number of ancestors, node-degrees, or the number of paths through a random or the j-th edge) could be treated in a similar way. By using analytic combinatorics techniques (see [7]) we obtain limiting distribution results for D n and L n (thus answering questions left open in [12,13]), whereas for the random variable (r.v. for short) P n (whose distributional treatment seems to be considerably more involved) we are able to give asymptotic results for the expectation.…”

On the Combinatorics of Binary Series-Parallel Graphs

“…Some properties, such as height, are strikingly different under the two probability measures. Here we use a similar approach: One of us, in [3], computed some parameters in such structures, when each bsp graph edge-addition sequence is considered equally likely. In the sections below we take the complementary point of view: Each of the C n−1 bsp graphs of size n is equally likely.…”

“…We study some combinatorics of directed binary series-parallel (bsp) graphs under the natural probability assumption: Distinct graphs of the same size are assigned the same likelihood. This provides an alternative model of randomness on bsp graphs to that in the incremental model of [3], where the author uses the same graph creation process, and assigns the same likelihood to each sequence of edge selections, even though many such different sequences lead to the same bsp graph. Resulting graphs of the same size have then a range of different probabilities.…”