Geographical database systems deal with certain basic topological relations such as "A overlaps B " and "B contains C " between simply connected regions in the plane. It is of great interest to make sound inferences from elementary statements of this form. This problem has been identified extensively in the recent literature, but very limited progress has been made towards addressing the considerable technical difficulties involved. In this paper we study the computational problems involved in developing such an inference system. We point out that the problem has two distinct components that interact in rather complex ways: relational consistency, and planarity. We develop polynomial-time algorithms for several important special cases, and prove almost all the others to be NP-hard. 1

Canonical models are very useful for determining simple representation formalism for qualitative relations. Allen's interval relations, e.g., can thereby be represented using the start and the end point of the intervals. Such a simple representation was not possible for regions of higher dimension as used by the Region Connection Calculus. In this paper we present a canonical model which allows regions and relations between them to be represented as points of the topological space and information about their neighbourhoods. With this formalism we are able to prove that whenever a set of RCC-8 formulas is consistent there exists a realization in any dimension, even when the regions are constrained to be (sets of) polytopes. For three- and higher dimensional space this is also true for internally connected regions. Using the canonical model we give algorithms for generating consistent scenarios. 1 Introduction The Region Connection Calculus (RCC) is a topological approach t...

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, and an O(n³)-time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point.

A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph (Pach and Tóth, 2001; Schaefer and ˇ Stefankovič, 2001). These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvíl showed that the recognition problem is NP-hard (Kratochvíl, 1991b). The result has consequences for the computational complexity of problems in graph drawing, and topological inference. We also show that the string graph problem is decidable for surfaces of arbitrary genus. Key words: String graphs, NP-completeness, graph drawing, topological inference, Euler diagrams

We introduce and study a modified notion of planarity, in which two regions of a map are considered adjacent when they share any point of their boundaries (not an edge, as standard planarity requires). We seek to characterize the abstract graphs realized by such map adjacencies. We prove some preliminary properties of such graphs, and give a polynomial time algorithm for the following restricted problem: given an abstract graph, decide whether it is realized by a map in which at most four regions meet at any point. The general recognition problem remains open.

. This paper examines the problem of testing consistency of sets of topological relations which are instances of the RCC-8 relation set (Randell, Cui and Cohn 1992). Representations of these relations as constraints within a number of logical frameworks are considered. It is shown that, if the arguments of the relations are interpreted as non-empty open sets within an arbitrary topological space, a complete consistency checking procedure can be provided by means of a composition table. This result is contrasted with the case where regions are required to be planar and bounded by Jordan curves, for which the consistency problem is known to be NP-hard. In order to investigate the completeness of compositional reasoning, the notion of k-compactness of a set of relations w.r.t. a theory is introduced. This enables certain consistency properties of relational networks to be examined independently of any specific interpretation of the domain of entities constrained by the relations. 1. Int...

Foundation (JU204/10-1). Abstract. We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open. 1

. Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced and their properties and inclusions between them are studied. Also the relation between planar and contact graphs is mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NP--complete (NP--hard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applic...

Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced, their properties and inclusions between them are studied, and the maximal clique in relation with the chromatic number of contact graphs is considered. Also relations between planar and contact graphs are mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NP--complete (NP--hard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applications in biology, see [14],[1]. We may also mention other kinds of intersection graphs s...

. Contact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are introduced and the inclusions between them are described, also the recognition of the contact graphs is studied. As one of the main results, it is proved that the recognition of 3--contact graphs is NP--complete for planar graphs, while the same question for planar triangulations is polynomial. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs, see [15],[1]; we may also mention other kinds ...