Crossing Number of Abstract Topological Graphs

We revoke the problem of drawing graphs in the plane so that only certain specified pairs of edges are allowed to cross. We overview some previous results and open problems, namely the connection to intersection graphs of curves in the plane. We complement these by stating a new conjecture and showing that its proof would solve the problem of algorithmic decidability of recognition of string graphs as well as realizability of abstract topological graphs and feasible drawability of graphs with restricted edge crossings. 1 Drawing Graphs When Only Some Pairs of Egdes Are Allowed to Cross Minimizing the number of crossing points in planar drawings of graphs in the plane is an important task. Too many crossing points – too dense a drawing – makes the chart badly readable for human eyes. It is well known that finding the exact minimum number of crossing points needed for planar drawing of a given graph, the so called crossing number, is an NP-hard problem [1]. On the other hand, this problem was deeply studied and well understood for many special graph classes and many exact results or at least tight bounds are known [6]. Much less studied is the question of drawing graphs for which only some pairs of egdes are allowed to cross. This natural generalization of planar graphs on one hand and crossing numbers on the other one was introduced in [4]. Practical motivation for this question are e.g. VLSI layouts with further restrictions – edges of certain types may not be allowed to cross because of their actual realization by connectors made from certain materials with different properties. Graph theoretical and complexity consequences of the concept of ‘allowing only certain pairs of edges to cross’ may bee even more interesting. Among them is the astonishing and challenging fact that so far there is no recursive algorithm known to decide the existence of a feasible drawing in this sense. In this note we want to reattract attention to the question of allowed crossings. We briefly review the definitions and give an overview of known results ? The author acknowledges partial support from Czech Research Grants GAUK 193 and 194 and GAČR 201/1996/0194, and Czech-US Science and Technology Research grant No. 94051. Mailing address: KAM MFF UK, Malostranské nám. 25, 118 00 Praha 1, Czech Republic. S.H. Whitesides (Ed.): GD’98, LNCS 1547, pp. 238–245, 1998. c © Springer-Verlag Berlin Heidelberg 1998 Crossing Number of Abstract Topological Graphs 239 and open problems. In the last section we report on a moderate progress in the question of recognition. Namely we introduce a new conjecture which, if settled in affirmative, would imply an exponential upper bound on the crossing number, and hence the existence of an (exponential but) finite recognition algorithm. A topological graph is a graph drawn in the plane with any number of edge crossings. However, some basic constraints are assumed, such as drawings of edges do not pass through vertices, any two edges share only a finite number of common points and they cross (i.e., do not touch) in the neighborhood of every common inner point (thus called a crossing point). No three edges pass through the same crossing point. A topological graph is thus a pair (G, D) where G is an abstract graph (i.e., G = (V, E) where V is a finite set of vertices and E ⊂ V2 ) is a set of edges, we consider undirected graphs without loops or multiple edges) and D is the drawing of G (D is a mapping from V ∪ E into points and Jordan arcs in the plane such that D(v) is the point representing vertex v in the drawing D and D(e) is the arc representing edge e, note that the arcs representing edges are considered open, i.e., their endpoints do not belong to the arcs). The drawing determines the set of crossing pairs of edges RD = {{e, f} : D(e) ∩ D(f) 6= ∅}. The first natural question is to reverse the reasoning. A pair (G, R) is called an abstract topological graph (briefly an AT-graph) if G is a graph and R ⊂ E2 ) is a set of pairs of its edges. (We use this notion as only crossing pairs of edges are specified, but not the actual drawing.) Then it is natural to ask if G allows a drawing D such that RD = R. Such a drawing (if it exists) is called a realization of (G, R). It is proved in [3] that it is NP-hard to decide if a given AT-graph is realizable in this sense. Though this approach has applications (e.g. for string graphs, see the next section), from the Graph Drawing point of view it seems more natural to ask whether (G, R) has a drawing D such that RD ⊆ R, i.e., whether G can be drawn in the plane so that only pairs of edges listed in R are allowed to cross. In [4], the AT-graph is called weak realizable if such a drawing exists. Let us call such a drawing feasible in this note. Obviously not every AT-graph has a feasible drawing, e.g., if R = ∅ then (G, ∅) has a feasible drawing if and only if G is planar. However, unlike planar graphs, it is NP-hard to decide if a given AT-graph allows a feasible drawing [3]. But the situation is even more serious than NP-completeness. So far no finite algorithm is known to decide realizability and weak realizability of AT-graphs, and it is even known that straightforward strategy to place the decision problem in the class NP fails (see the next section). In some sense this makes the problem theoretically more interesting, since there are not many graph theoretical problems with practical motivation which float above the class NP. (Perhaps it is the topological nature of the problem that causes this effect. Another problem for which no finite algorithm was known for a while, are linklessly embedabble graphs, now polynomial by Robertson-Seymour Graph Minor Machinery.)