Egon Wanke

k -NLC graphs and polynomial algorithms

Abstract We introduce the class of k-node label controlled (NLC) graphs and the class of k-NLC trees. Each k-NLC graph is an undirected tree-structured graph, where k is a positive integer. The class of k-NLC trees is a proper subset of the class of k-NLC graphs. Both classes include many interesting graph families. For instance, each partial k-tree is a (2k + 1 − 1)-NLC tree and each co-graph is a 1-NLC graph. Furthermore, we introduce a very general method for the design of polynomial algorithms for NP-complete graph problems, where the input graphs are restricted to tree-structured graphs. We exemplify our method with the SIMPLE MAX-CUT PROBLEM and the HAMILTONIAN CIRCUIT PROPERTY on k-NLC graphs.

An Algorithm for Finding Maximal Common Subtopologies in a Set of Protein Structures

For the comparison and analysis of protein structures, it is of interest to find maximal common substructures in a given set of proteins. This question is also relevant for motif definition and structure classification. In this paper we describe first a new suitable representation of the secondary structure topology of a protein by an undirected labeled graph. Based on this representation we developed a new fast algorithm that finds all common subtopologies in a set of protein structures. Our method is based on the algorithm by Bron and Kerbosch (1973), which enumerates all maximal cliques in a graph. The main improvement of our algorithm is to restrict the search process to cliques that represent connected substructures. This restriction reduces the number of cliques to be considered during the search process and the size of the search tree drastically. Thus we are able to handle large proteins. Experiments show the efficiency and superiority of our algorithm in comparison with other existing algorithms basing on graph-theoretical methods.

How to Solve NP-hard Graph Problems on Clique-Width Bounded Graphs in Polynomial Time

We show that many non-MSO1 NP-hard graph problems can be solved in polynomial time on clique-width and NLC-width bounded graphs using a very general and simple scheme. Our examples are partition into cliques, partition into triangles, partition into complete bipartite subgraphs, partition into perfect matchings, partition into forests, cubic subgraph, Hamiltonian path, minimum maximal matching, and vertex/edge separation problems.

Mapping brains without coordinates

Brain mapping has evolved considerably over the last century. While most emphasis has been placed on coordinate-based spatial atlases, coordinate-independent parcellation-based mapping is an important technique for accessing the multitude of structural and functional data that have been reported from invasive experiments, and provides for flexible and efficient representations of information. Here, we provide an introduction to motivations, concepts, techniques and implications of coordinate-independent mapping of microstructurally or functionally defined brain structures. In particular, we explain the problems of constructing mapping paths and finding adequate heuristics for their evaluation. We then introduce the three auxiliary concepts of acronym-based mapping (AM), of a generalized hierarchy (GM ontology), and of a topographically oriented regional map (RM) with adequate granularity for mapping between individual brains with different cortical folding and between humans and non-human primates. Examples from the CoCoMac database of primate brain connectivity demonstrate how these concepts enhance coordinate-independent mapping based on published relational statements. Finally, we discuss the strengths and weaknesses of spatial coordinate-based versus coordinate-independent microstructural brain mapping and show perspectives for a wider application of parcellation-based approaches in the integration of multi-modal structural, functional and clinical data.

The Tree-Width of Clique-Width Bounded Graphs Without Kn, n

We proof that every graph of clique-width k which does not contain the complete bipartite graph Kn,n for some n > 1 as a subgraph has tree-width at most 3k(n - 1) - 1. This immediately implies that a set of graphs of bounded clique-width has bounded tree-width if it is uniformly l-sparse, closed under subgraphs, of bounded degree, or planar.

Efficient solution to connectivity problems on hierarchically defined graphs

Using hierarchical definitions we can describe very large graphs in small spaces. In this paper we discuss how such succinct graph descriptions can be used to speed up the solution of graph problems. We present the bottom-up method that solves graph problems without expanding the hierarchical description. This allows solutions that are efficient in terms of the hierarchical graph description instead of the size of the expanded graph. We exemplify the method by giving efficient solutions to connectivity problems on hierarchical graphs. Our results have applications in computer-aided design for integrated circuit design and other engineering problems.

Efficient analysis of graph properties on context-free graph languages

We consider efficient ways of analyzing families of hierarchical engineering designs, using methods from the area of graph grammars. Our approach starts from an equivalent reformulation of hyperedge replacement systems that is particularly well suited for complexity analysis. We define a characteristic called ”finiteness” of graph properties and give a combinatorial decision algorithm for deciding whether a graph language generated by a given cellular graph grammar has a graph with a given finite graph property. We introduce structural parameters that bound the complexity of the decision procedure and discuss special cases for which the decision can be made in polynomial time. Extensions to non context-free graph grammars are also given. Our results provide explicit and efficient combinatorial algorithms solving problems whose decidability has been proved in a general framework by Courcelle.

On the relationship between NLC-width and linear NLC-width

In this paper, we consider NLC-width, NLCT-width, and linear NLC-width bounded graphs. We show that the set of all complete binary trees has unbounded linear NLC-width and that the set of all co-graphs has unbounded NLCT-width. Since trees have NLCT-width 3 and co-graphs have NLC-width 1, it follows that the family of linear NLC-width bounded graph classes is a proper subfamily of the family of NLCT-width bounded graph classes and that the family of NLCT-width bounded graph classes is a proper subfamily of the family of NLC-width bounded graph classes.

2008 Special Issue: Deducing logical relationships between spatially registered cortical parcellations under conditions of uncertainty

We propose a new technique, called Spatial Objective Relational Transformation (SORT), as an automated approach for derivation of logical relationships between cortical areas in different brain maps registered in the same Euclidean space. Recently, there have been large amounts of voxel-based three-dimensional structural and functional imaging data that provide us with coordinate-based information about the location of differently defined areas in the brain, whereas coordinate-independent, parcellation-based mapping is still commonly used in the majority of animal tracing and mapping studies. Because of the impact of voxel-based imaging methods and the need to attribute their features to coordinate-independent brain entities, this mapping becomes increasingly important. Our motivation here is not to make vague statements where more precise spatial statements would be better, but to find criteria for the identity (or other logical relationships) between areas that were delineated by different methods, in different individuals, or mapped to three-dimensional space using different deformation algorithms. The relevance of this problem becomes immediately obvious as one superimposes and compares different datasets in multimodal databases (e.g. CARET, http://brainmap.wustl.edu/caret), where voxel-based data are registered to surface nodes exploited by the procedure presented here. We describe the SORT algorithm and its implementation in the Java 2 programming language (http://java.sun.com/, which we make available for download. We give an example of practical use of our approach, and validate the SORT approach against a database of the coordinate-independent statements and inferences that have been deduced using alternative techniques.

Minimum Cost Paths in Periodic Graphs

We consider graphs with