A clique-difference encoding scheme for labelled k-path graphs

AbstractWe present in this paper a codeword for labelled k-path graphs. Structural properties of this codeword are investigated, leading to the solution of two important problems: determining the exact number of labelled k-path graphs with n vertices and locating a hamiltonian path in a given k-path graph in time O(n). The corresponding encoding scheme is also presented, providing linear-time algorithms for encoding and decoding

The Complexity of Bisimulation and Simulation on Finite Systems

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and S\'antha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width

On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids

Ackermann Encoding, Bisimulations, and OBDDs

We propose an alternative way to represent graphs via OBDDs based on the observation that a partition of the graph nodes allows sharing among the employed OBDDs. In the second part of the paper we present a method to compute at the same time the quotient w.r.t. the maximum bisimulation and the OBDD representation of a given graph. The proposed computation is based on an OBDD-rewriting of the notion of Ackermann encoding of hereditarily finite sets into natural numbers.Comment: To appear on 'Theory and Practice of Logic Programming

Algorithmic Meta-Theorems

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This paper gives a survey of algorithmic meta-theorems obtained in recent years and the methods used to prove them. As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic consequences of this theory as far as they are needed in the theory of algorithmic meta-theorems

A survey of frequent subgraph mining algorithms

AbstractGraph mining is an important research area within the domain of data mining. The field of study concentrates on the identification of frequent subgraphs within graph data sets. The research goals are directed at: (i) effective mechanisms for generating candidate subgraphs (without generating duplicates) and (ii) how best to process the generated candidate subgraphs so as to identify the desired frequent subgraphs in a way that is computationally efficient and procedurally effective. This paper presents a survey of current research in the field of frequent subgraph mining and proposes solutions to address the main research issues.</jats:p

Monadic second-order definable graph orderings

We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic second-order logic, possibly with the help of monadic parameters. We consider two variants of monadic second-order logic: one where we can only quantify over sets of vertices and one where we can also quantify over sets of edges. For several special cases, we present combinatorial characterisations of when such a linear ordering is definable. In some cases, for instance for graph classes that omit a fixed graph as a minor, the presented conditions are necessary and sufficient; in other cases, they are only necessary. Other graph classes we consider include complete bipartite graphs, split graphs, chordal graphs, and cographs. We prove that orderability is decidable for the so called HR-equational classes of graphs, which are described by equation systems and generalize the context-free languages

On coding labeled trees

Trees are probably the most studied class of graphs in Computer Science. In this thesis we study bijective codes that represent labeled trees by means of string of node labels. We contribute to the understanding of their algorithmic tractability, their properties, and their applications. The thesis is divided into two parts. In the first part we focus on two types of tree codes, namely Prufer-like codes and Transformation codes. We study optimal encoding and decoding algorithms, both in a sequential and in a parallel setting. We propose a unified approach that works for all Prufer-like codes and a more generic scheme based on the transformation of a tree into a functional digraph suitable for all bijective codes. Our results in this area close a variety of open problems. We also consider possible applications of tree encodings, discussing how to exploit these codes in Genetic Algorithms and in the generation of random trees. Moreover, we introduce a modified version of a known code that, in Genetic Algorithms, outperform all the other known codes. In the second part of the thesis we focus on two possible generalizations of our work. We first take into account the classes of k-trees and k-arch graphs (both superclasses of trees): we study bijective codes for this classes of graphs and their algorithmic feasibility. Then, we shift our attention to Informative Labeling Schemes. In this context labels are no longer considered as simple unique node identifiers, they rather convey information useful to achieve efficient computations on the tree. We exploit this idea to design a concurrent data structure for the lowest common ancestor problem on dynamic trees. We also present an experimental comparison between our labeling scheme and the one proposed by Peleg for static trees

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm