Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft
Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a given number of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs with a given minimum degree, and establishes new ones, such as the enumeration of Euler graphs, i.e. where all vertices have an even degree. Those results are derived using analytic combinatorics.
We provide an algorithm that, given a tree homomorphism H and a regular tree language L represented by a tree automaton, determines whether H(L) is regular. This settles a question that has been open for a long time.Along the way, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular tree languages through tree homomorphisms.Our algorithms are based on the following constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with inequality constraints recognizing the complementary language. We also define a new class of automata with arbitrary inequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with inequality constraints and the image of a regular tree language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism.
The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and seriesparallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].
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