Treewidth-Two Graphs as a Free Algebra

We give a new and elementary proof that the graphs of treewidth at most two can be seen as a free algebra. This result was originally established through an elaborate analysis of the structure of K_4-free graphs, ultimately reproving the well-known fact that the graphs of treewidth at most two are precisely those excluding K_4 as a minor. Our new proof is based on a confluent and terminating rewriting system for term-labeled graphs and does not involve graph minors anymore. The new strategy is simpler and robust in the sense that it can be adapted to subclasses of treewidth-two graphs, e.g., graphs without self-loops

K4-free graphs as a free algebra

Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra,answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition to parallel composition and series composition, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equationa lpresentation and we prove it complete: two terms from the syntax are congruent if and only if they denote the same graph

Datalog and Constraint Satisfaction with Infinite Templates

On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that appeared at STACS'06. In the third version in the arxiv we have revised the presentation again and added a section that relates our results to formalizations of CSPs using relation algebra

Benchmarks for Parity Games (extended version)

We propose a benchmark suite for parity games that includes all benchmarks that have been used in the literature, and make it available online. We give an overview of the parity games, including a description of how they have been generated. We also describe structural properties of parity games, and using these properties we show that our benchmarks are representative. With this work we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from https://github.com/jkeiren/paritygame-generator. This is an extended version of the paper that has been accepted for FSEN 201

On giant components and treewidth in the layers model

Given an undirected $n$-vertex graph $G(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex induced subgraph of $G$ generated by ordering $V$ according to a random permutation $\pi$ and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} $k$. The layers model has found applications in studying $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are $3$-regular graphs $G$ for which with high probability $T_3(G)$ has a connected component of size $\Omega(n)$. Moreover, this connected component has treewidth $\Omega(n)$. This lower bound on the treewidth extends to many other random graph models. In contrast, $T_2(G)$ is known to be a forest (hence of treewidth~1), and we establish that if $G$ is of bounded degree then with high probability the largest connected component in $T_2(G)$ is of size $O(\log n)$. We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability $1$