Characterisations and Examples of
Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties.
In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed , there exists a class with bounded expansion, such that a random graph of order and edge probability asymptotically almost surely belongs to the class.
We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with ‘linear’ crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.
Key words and phrases:graph, queue layout, queue-number, stack layout, stack-number, book embedding, book thickness, page-number, expansion, bounded expansion, crossing number, non-repetitive chromatic number, Thue number, random graph, jump number
1991 Mathematics Subject Classification:05C62 (graph representations), 05C15 (graph coloring), 05C83 (graph minors)
What is a ‘sparse’ graph? It is not enough to simply consider edge density as the measure of sparseness. For example, if we start with a dense graph (even a complete graph) and subdivide each edge by inserting a new vertex, then in the obtained graph the number of edges is less than twice the number of vertices. Yet in several aspects, the new graph inherits the structure of the original.
A natural restriction is to consider proper minor-closed graph classes. These are the classes of graphs that are closed under vertex deletions, edge deletions, and edge contractions (and some graph is not in the class). Planar graphs are a classical example. Interest in minor-closed classes is widespread. Most notably, RS-GraphMinors proved that every minor-closed class is characterised by a finite set of excluded minors. (For example, a graph is planar if and only if it has no -minor and no -minor.) Moreover, membership in a particular minor-closed class can be tested in polynomial time. There are some limitations however in using minor-closed classes as models for sparse graphs. For example, cloning each vertex (and its incident edges) does not preserve such properties. In particular, the graph obtained by cloning each vertex in the planar grid graph has unbounded clique minors [Wood-ProductMinor].
A more general framework concerns proper topologically-closed classes of graphs. These classes are characterised as follows: whenever a subdivision of a graph belongs to the class then also belongs to the class (and some graph is not in the class). Such a class is characterised by a possibly infinite set of forbidden configurations.
A further generalisation consists in classes of graphs having bounded expansion, as introduced by ICGT05, Taxi_stoc06, NesOdM-GradI. Roughly speaking, these classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Thus bounded expansion classes are broader than minor-closed classes, which are those classes for which every minor of every graph in the class has bounded average degree.
Bounded expansion classes have a number of desirable properties. (For an extensive study we refer the reader to [NesOdM-GradI, NesOdM-GradII, NesOdM-GradIII, Dvo, Dvorak-EUJC08].) For example, they admit so-called low tree-depth decompositions [NesOdM-TreeDepth-EJC06], which extend the low tree-width decompositions introduced by DDOSRSV-JCTB04 for minor-closed classes. These decompositions, which may be computed in linear time, are at the core of several linear-time graph algorithms, such as testing for an induced subgraph isomorphic to a fixed pattern [Taxi_stoc06, NesOdM-GradII]. In fact, isomorphs of a fixed pattern graph can be counted in a graph from a bounded expansion class in linear time [Taxi_hom]. Also, low tree-depth decompositions imply the existence of restricted homomorphism dualities for classes with bounded expansion [NesOdM-GradIII]. That is, for every class with bounded expansion and every connected graph (which is not necessarily in ) there exists a graph such that
where means that there is a homomorphism from to , and means that there is no such homomorphism. Finally, note that the structural properties of bounded expansion classes make them particularly interesting as a model in the study of ‘real-world’ sparse networks [Aeolus06].
Bounded expansion classes are the focus of this paper. Our contributions to this topic are classified as follows (see Figure LABEL:fig:bec):
We establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts; see Section LABEL:sec:Characterisations.
This latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree (that is, for random graphs of order with edge probability ); see Section LABEL:sec:Random.
We present several new examples of classes with bounded expansion that appear naturally in the context of graph drawing or graph colouring. In particular, we prove that each of the following classes have bounded expansion, even though they are not contained in a proper topologically-closed class:
graphs that can be drawn with a bounded number of crossings per edge (Section LABEL:sec:CrossingNumber),
graphs with bounded queue-number (Section LABEL:sec:Queue),
graphs with bounded stack-number (Section LABEL:sec:Stack),
graphs with bounded non-repetitive chromatic number (Section LABEL:sec:NonRep).
We also prove that graphs with ‘linear’ crossing number are contained in a topologically-closed class, and graphs with bounded crossing number are contained in a minor-closed class (Section LABEL:sec:CrossingNumber).