We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in series-parallel graphs (and related graph classes) with n vertices. In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988]. We also show that these two parameters converge to a central limit law.

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k . Our technique applies to general families of problems where standard dynamic programming runs in 2 O ( k ⋅log k ) ⋅ n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called surface cut decomposition , generalizing sphere cut decompositions of planar graphs, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of noncrossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k ) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2 O ( k ) ⋅ n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.

We present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works in both the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp., $g_n$) of labelled (resp., unlabelled) graphs on n vertices from a subcritical graph class ${\mathcal{G}}=\cup_n {\mathcal{G}_n}$ satisfies asymptotically the universal behavior $g_n = c \!n^{-5/2} \!\gamma^n \! (1+o(1))$ for computable constants $c,\gamma$, e.g., $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree k (k fixed) in a graph chosen uniformly at random from $\mathcal{G}_n$ converges (after rescaling) to a normal law as $n\to\infty$.