A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mn "ev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.
The recognition problem for point visibility graphs has been explicitly stated as an important open problem by various authors @cite_13 , and is listed as the first open problem in a recent survey from Ghosh and Goswami @cite_20 .
@cite_13 @cite_20
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In this paper, we present two necessary conditions for recognizing point visibility graphs. We show that this recognition problem lies in PSPACE. We state new properties of point visibility graphs along with some known properties that are important in understanding point visibility graphs. For planar point visibility graphs, we present a complete characterization which leads to a linear time recognition and reconstruction algorithm. This manuscript introduces the problem of prominent object detection and recognition inspired by the fact that human seems to priorities perception of scene elements. The problem deals with finding the most important region of interest, segmenting the relevant item object in that area, and assigning it an object class label. In other words, we are solving the three problems of saliency modeling, saliency detection, and object recognition under one umbrella. The motivation behind such a problem formulation is (1) the benefits to the knowledge representation-based vision pipelines, and (2) the potential improvements in emulating bio-inspired vision systems by solving these three problems together. We are foreseeing extending this problem formulation to fully semantically segmented scenes with instance object priority for high-level inferences in various applications including assistive vision. Along with a new problem definition, we also propose a method to achieve such a task. The proposed model predicts the most important area in the image, segments the associated objects, and labels them. The proposed problem and method are evaluated against human fixations, annotated segmentation masks, and object class categories. We define a chance level for each of the evaluation criterion to compare the proposed algorithm with. Despite the good performance of the proposed baseline, the overall evaluations indicate that the problem of prominent object detection and recognition is a challenging task that is still worth investigating further. Given a 3-SAT formula, a graph can be constructed in polynomial time such that the graph is a point visibility graph if and only if the 3-SAT formula is satisfiable. This reduction establishes that the problem of recognition of point visibility graphs is NP-hard. A general theorem on the complexity of a class of recognition problems is proved. As a particular case the following result is given: There is no algorithm which, for any 2-coloration of the infinite complete graph, can produce a monochromatic subgraph of k vertices within 2k2 steps (at each step the color of an arbitrary edge is questioned).