Multigraphs without large bonds are wqo by contraction

We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental

A General Framework for Well-Structured Graph Transformation Systems

Graph transformation systems (GTSs) can be seen as wellstructured transition systems (WSTSs), thus obtaining decidability results for certain classes of GTSs. In earlier work it was shown that wellstructuredness can be obtained using the minor ordering as a well-quasiorder. In this paper we extend this idea to obtain a general framework in which several types of GTSs can be seen as (restricted) WSTSs. We instantiate this framework with the subgraph ordering and the induced subgraph ordering and apply it to analyse a simple access rights management system.Comment: Extended version (including proofs) of a paper accepted at CONCUR 201

Hypertree-depth and minors in hypergraphs

AbstractWe introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs. We characterise hypergraphs of bounded hypertree-depth by the ultramonotone robber and marshals game, by vertex-hyperrankings and by centred hypercolourings.Furthermore, we show that minors in hypergraphs are ‘well-behaved’ with respect to hypertree-depth and other hypergraph invariants, such as generalised hypertree-depth and generalised hyperpath-width.We work in the framework of hypergraph pairs (G,H), consisting of a graph G and a hypergraph H that share the same vertex set. This general framework allows us to obtain hypergraph minors, graph minors and induced graph minors as special cases

Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, Eta Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP is in coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.Comment: To appear in the Journal of Computer and System Science

Well-quasi-orders in subclasses of bounded treewidth graphs

We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded feedback-vertex-set are well-quasi-ordered by the topological-minor order, graphs with bounded vertex-covers are well-quasi-ordered by the subgraph order, and graphs with bounded circumference are well-quasi-ordered by the induced-minor order. Our results give an algorithm for recognizing any graph family in these classes which is closed under the corresponding minor order refinement

Verification of Well-Structured Graph Transformation Systems

The aim of this thesis is the definition of a high-level framework for verifying concurrent and distributed systems. Verification in computer science is challenging, since models that are sufficiently expressive to describe real-life case studies suffer from the undecidability of interesting problems. This also holds for the graph transformation systems used in this thesis. To still be able to analyse these system we have to restrict either the class of systems we can model, the class of states we can express or the properties we can verify. In fact, in the framework we will present, all these limitations are possible and each allows to solve different verification problems. For modelling we use graphs as the states of the system and graph transformation rules to model state changes. More precisely, we use hypergraphs, where an edge may be incident to an arbitrary long sequence of nodes. As rule formalism we use the single pushout approach based on category theory. This provides us with a powerful formalisms that allows us to use a finite set of rules to describe an infinite transition system. To obtain decidability results while still maintaining an infinite state space we use the theory of well-structured transition systems (WSTS), the main source of decidability results in the infinite case. We need to equip our state space with a well-quasi-order (wqo) which is a simulation relation for the transition relation (this is also known as compatibility condition or monotonicity requirement). If a system can be seen as a WSTS and some additional conditions are satisfied, one can decide the coverability problem, i.e., the problem of verifying whether, from a given initial state one can reach a state that covers a final state, i.e. is larger than the final state with respect to a chosen order. This problem can be used for verification by giving a finite set of minimal error states that represent an infinite class of erroneous states (i.e. all larger states). By checking whether one of these minimal states is coverable, we verify whether an error is reachable. The theory of WSTS provides us with a generic backwards algorithm to solve this problem. For graphs we will introduce three orders, the minor ordering, the subgraph ordering and the induced subgraph ordering, and investigate which graph transformation systems form WSTS with these orders. Since only the minor ordering is a wqo on all graphs, we will first define so-called Q-restricted WSTS, where we only require that the chosen order is a wqo on the downward-closed class Q. We examine how this affects the decidability of the coverability problem and present appropriate classes Q such that the subgraph ordering and induced subgraph ordering form Q-restricted WSTS. Furthermore, we will prove the computability of the backward algorithm for these Q-restricted WSTS. More precisely, we will do this in the form of a framework and give necessary conditions for orders to be compatible with this framework. For the three mentioned orders we prove that they satisfy these conditions. Being compatible with different orders strengthens the framework in the following way: On the one hand error specifications have to be invariant wrt. the order, meaning that different orders can describe different properties. On the other hand, there is the following trade-off: coarser orders are wqos on larger sets of graphs, but fewer GTS are well-structured wrt. coarse orders (analogously the reverse holds for fine orders). Finally, we will present the tool Uncover which implements most of the theoretical framework defined in this thesis. The practical value of our approach is illustrated by several case studies and runtime results

Well-Quasi-Orders in Subclasses of Bounded Treewidth Graphs and their Algorithmic Applications

We show that three subclasses of bounded treewidth graphs are well-quasi-ordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertex-covers are well quasi ordered by the induced subgraph order, graphs with bounded feedback-vertex-set are well quasi ordered by the topological-minor order, and graphs with bounded circumference are well quasi ordered by the induced-minor order. Our results give algorithms for recognizing any graph family in these classes which is closed under the corresponding minor order refinement