Enumerating and Generating Labeled k-degenerate Graphs

Abstract

A k-degenerate graph is a graph in which every induced subgraph has a vertex with degree at most k. The class of k-degenerate graphs is interesting from a theoretical point of view and it plays an interesting role in the theory of fixed parameter tractability since some otherwise W[2]-hard domination problems become fixed-parameter tractable for k-degenerate graphs.It is a well-known fact that the k-degenerate graphs are exactly the graphs whose vertex-set can be well-ordered such that each vertex is incident to at most k larger vertices with respect to this ordering. A well-ordered k-degenerate graph is a labeled graph with vertex-labels 1, . . . , n such that the ordering of the vertices by their labels is a well-ordering of the graph.We consider the problem of enumerating and generating well-ordered k-degenerate graphs with a given number of vertices and with a given number of vertices and edges, respectively, uniformly at random. By generating wellordered k-degenerate graphs we generate at least one labeled copy of each unlabeled k-degenerate graph and we filter some but not all isomorphies compared to the classical labeled approach.We also introduce the class of strongly k-degenerate graphs, which are k-degenerate graphs with minimum degree k. These graphs are a natural generalization of k-regular graphs which can be used in order to generate graphs with predefined core-decomposition.We present efficient algorithms for generating wellordered k-degenerate graphs with given number of vertices (and edges). After a precomputation which must only be performed once when generating more than one well-ordered k-degenerate graph these algorithms are almost optimal. Additionally, we present complete non-uniform generators for these classes with optimal running time. We also present an efficient and complete generator for well-ordered strongly k-degenerate graphs with given number of vertices (and edges). Finally, we present efficient algorithms for enumerating well-ordered k-degenerate and strongly k-degenerate graphs.

Fixed points of graph peeling

“…The subgraph determined by brown edges corresponds for example to a fixed point of peeling value 7. We denote by F P k the class of graphs that are fixed points of degree peeling k. They are also called strongly k-degenerate graphs in [17]. F P k includes well-known classes of graphs.…”

“…The upper bound is the number of edges in a edge-maximal F P k graphs with n vertices i.e. graphs such that an edge can not be added between two independent vertices without increasing the maximum peeling value [17]. Graphs generated using the Barabási-Albert model [18] with a clique of size k as seed for example are in this case, edge maximal F P k .…”

Greedy Algorithms, Ordering of Variables, and d-degenerate Instances

“…It turns out that sampling random d-degenerate formulas is much more difficult than sampling just random formulas, and even defining a proper distribution is trickier. We use the dynamic programming algorithm for generating ddegenerate graphs from [3], and amend it so that it outputs 2-CNFs. Distribution Φ d−gen (n, pn, d), where n is the number of variables, p is the clause-to-variables ratio, and d is the degeneracy parameter, is defined as follows: We fix an order of variables x 1 , .…”

“…In the case of random d-degenerate formulas let D(n, m, d) denote the number of d-degenerate formulas with n variables and m clauses. These numbers are byproducts of the random formula generating algorithm from [3]. Then…”

External-Memory Network Analysis Algorithms for Naturally Sparse Graphs

“…In the RAM model, this greedy algorithm takes O(n) time (e.g., see [5]). Bauer et al [6] describe methods for generating such graphs and their d-degeneracy orderings at random.…”