Unsolved Problems in Visibility Graph Theory
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چکیده
The visibility graph is a fundamental structure studied in the field of computational geometry, and pose some special challenges [12, 26]. Apart from theoretical interests, visibility graphs has important applications also. Some of the early applications include computing Euclidean shortest paths in the presence of obstacles [36] and decomposing two-dimensional shapes into clusters [40]. For more on the uses of this class of graphs, see [37, 43]. Let P be a simple polygon with or without holes in the plane. We say two points a and b in P are mutually visible if the line segment ab lies entirely within P . This definition allows the segment ab to pass through a reflex vertex or graze along a polygonal edge. The visibility graph (also called the vertex visibility graph) G of P is defined by associating a node with each vertex of P such that (vi, vj) is an undirected edge of G if polygonal vertices vi and vj are mutually visible. Figure 1(b) shows the visibility graph of the polygon in Figure 1(a). Sometimes the visibility graph is drawn directly on the polygon, as shown in Figure 1(c). It can be seen that every triangulation of P corresponds to a subgraph of the visibility graph of P .
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تاریخ انتشار 2009