Complexity Aspects of Visibility Graphs

Abstract

In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove: • Θ(n3) area is necessary and sufficient to realize the complete visibility graph Kn. • There exist visibility graphs which require exponential area to realize. • Any maximal outerplanar graph of diameter d can be realized in O(d2 · 2d) area, which can be as small as O(n log2 n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n8) area. The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P1 and P2, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

Unsolved problems in visibility graphs of points, segments, and polygons

“…This is the only upper bound known on the complexity of the problem. Abello and Kumar [4] studied the relationship between visibility graphs and oriented matroids, Lin and Skiena [74] studied the equivalent order types, and Streinu [109,110] and O'Rourke and Streinu [87] studied psuedo-line arrangements. Everett and Corneil [43,45] have solved the reconstruction problem for the visibility graphs of spiral polygons and the corresponding problem for the visibility graph of tower polygons has been solved by Choi, Shin and Chwa [19].…”

“…A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge. The minimum dominating set problem in visibility graphs corresponds to the art gallery problem in polygons which has been shown to be NP-hard [72,74]. Following the approximation algorithm for the art gallery problem for polygons given by Ghosh [52,56], a minimum dominating set of visibility graph can be computed with an approximation ratio of O(log n).…”

Computing the maximum clique in the visibility graph of a simple polygon

“…Shermer [20] showed that the maximum independent set problem is also NP-hard. An independent set in the visibility graph is also called hidden set which has been studied by Eidenbenz [6,7], Ghosh et al [12] and Lin and Skiena [16]. Further, Lin and Skiena [16] showed that the problems of finding a minimum vertex cover and a maximum dominating set in the visibility graph of a simple polygon are NP-hard.…”

“…An independent set in the visibility graph is also called hidden set which has been studied by Eidenbenz [6,7], Ghosh et al [12] and Lin and Skiena [16]. Further, Lin and Skiena [16] showed that the problems of finding a minimum vertex cover and a maximum dominating set in the visibility graph of a simple polygon are NP-hard. Lin and Skiena [16] have shown that the problem of determining whether the visibility graphs of two simple polygons are isomorphic is isomorphic-complete.…”

A combinatorial view of visibility graphs of simple polygons

“…It is only known to be in PSPACE [8]. For additional work on the question of membership of this problem in N P see [15]. From the characterization standpoint, Ghosh [ll] obtained four necessary conditions for internal visibility graphs of simple polygons but it has been shown that they are not sufficient even for triconnected graphs [6, 81.…”