679 research outputs found
Grid-Obstacle Representations with Connections to Staircase Guarding
In this paper, we study grid-obstacle representations of graphs where we
assign grid-points to vertices and define obstacles such that an edge exists if
and only if an -monotone grid path connects the two endpoints without
hitting an obstacle or another vertex. It was previously argued that all planar
graphs have a grid-obstacle representation in 2D, and all graphs have a
grid-obstacle representation in 3D. In this paper, we show that such
constructions are possible with significantly smaller grid-size than previously
achieved. Then we study the variant where vertices are not blocking, and show
that then grid-obstacle representations exist for bipartite graphs. The latter
has applications in so-called staircase guarding of orthogonal polygons; using
our grid-obstacle representations, we show that staircase guarding is
\textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Some results on open edge guarding of polygons
This paper focuses on a variation of the Art Gallery problem that considers open edge guards. The “open” prefix means the endpoints of an edge where a guard is are not taken into account for visibility purposes. This paper studies the number of open edge guards that are sufficient and sometimes necessary to guard some classes of simple polygons.Fondo Europeo de Desarrollo RegionalEuropean Science FoundationMinisterio de Ciencia e InnovaciónFundação para a Ciência e a Tecnologi
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