Gregorio Hernández

Bipartite embedding of trees in the plane

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. Now imagine the set P is partitioned into two disjoint subsets R and B, and we ask for an embedding of T in P without crossings and with the property that all edges join a point in R (red) and a point in B (blue). In this case we say that T admits a bipartite embedding with respect to the bipartition (R, B). Examples show that the problem in its full generality is not solvable. In view of this fact we consider several embedding problems and study for which bipartitions they can be solved. We present several results that are valid for any bipartition (R, B) in general position, and some other results that hold for particular configurations of points.

A combinatorial property of convex sets

A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn, c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements Si, Sj of Φ such that any set S′ homothetic to S that contains them contains n/c elements of Φ, c a constant (S is homothetic to S if 5’ = λS + v, where λ is a real number greater than 0 and v is a vector of ℜ2). Our proof method is based on a new type of Voronoi diagram, called the “closest covered set diagram” based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in ℜ3)3 such that for any nonempty subset ΦHh of Φ there is a sphere SH containing all the elements of ΦH, and no other element of Φ.

Voronoi diagrams and containment of families of convex sets on the plane

It is known that any family Pn of n points on the plane contains two elements such that any circle containing them contains *elements of Pn. We prove: Let @be a family of n disjoint compact convex sets on the plane, S be a strictly convex compact set. Then there are two elements Si, Sj of @ such that any set S’ homothetic to S that contains them contains ~ elements of 0, c a constant. Our proof method is based on a new type of Voronoi diagram, called the “closest covered set diagram” based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Y of n disjoint convex sets in ‘33 such that for any subset H of Y there is a sphere containing all of the elements of H, and no other element of Y-H is contained in it. Finally, we show using closed covered Voronoi diagrams that the ordered set B5 consisting of five bottom elements and ten top elements, one above each pair of bottoms, is not a circle order.

Distance domination, guarding and covering of maximal outerplanar graphs

In this paper we introduce the notion of distance k -guarding applied to triangulation graphs, and associate it with distance k -domination and distance k -covering. We obtain results for maximal outerplanar graphs when k = 2 . A set S of vertices in a triangulation graph T is a distance 2-guarding set (or 2 d -guarding set for short) if every face of T has a vertex adjacent to a vertex of S . We show that ? n 5 ? (respectively, ? n 4 ? ) vertices are sufficient to 2 d -guard and 2 d -dominate (respectively, 2 d -cover) any n -vertex maximal outerplanar graph. We also show that these bounds are tight.

Estimating the Maximum Hidden Vertex Set in Polygons

It is known that the maximum hidden vertex set problem on a given simple polygon is NP-hard (Shermer, 1989), therefore we focused on the development of approximation algorithms to tackle it. We propose four strategies to solve this problem, the first two (based on greedy constructive search) are designed specifically to solve it, and the other two are based on the general metaheuristics simulated annealing and genetic algorithms. We conclude, through experimentation, that our best approximate algorithm is the one based on the Simulated Annealing metaheuristic. The solutions obtained with it are very satisfactory in the sense that they are always close to optimal (with an approximation ratio of 1.7, for arbitrary polygons; and with an approximation ratio of 1.5, for orthogonal polygons). We, also, conclude,that on average the maximum number of hidden vertices in a simple polygon (arbitrary or orthogonal) with n vertices is n/4.

Monitoring maximal outerplanar graphs

In this paper we define a new concept of monitoring the elements of triangulation graphs by faces. Furthermore, we analyze this, and other monitoring concepts (by vertices and by edges), from a combinatorial point of view, on maximal outerplanar graphs.

Metaheuristic Approaches for the Minimum Vertex Guard Problem

We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. This problem is NP-hard with relevant practical applications. In this paper we propose three metaheuristic approaches to this problem. Combined with the genetic algorithms strategy, which was proposed in [4], these four approximation algorithms have been implemented and compared. The experimental evaluation from the hybrid strategy shows significant improvement in the number of guards compared to theoretical bounds.

Solving the illumination problem with heuristics

In this article we propose optimal and quasi optimal solutions to the problem of searching for the maximum lighting point inside a polygon P of n vertices. This problem is solved by using three different techniques: random search, simulated annealing and gradient. Our comparative study shows that simulated annealing is very competitive in this application. To accomplish the study, a new polygon generator has been implemented, which greatly helps in the general validation of our claims on the illumination problem as a new class of optimization task.

Remote Monitoring by Edges and Faces of Maximal Outerplanar Graphs

In Graph Theory, the notion of monitoring vertices or edges of graphs by other vertices or edges has been widely studied (see, e.g. [1]). For instance, the well-known concepts of domination, edge domination, vertex covering and edge covering. In Computational Geometry, for triangulation graphs, i.e. a triangulation of a set of points in the plane, a different monitoring notion was established—the notion of monitoring bounded faces (faces, for short). When the faces are monitored by vertices or edges, we obtain the parameters associated with vertex guarding or edge guarding, respectively. The guarding concepts on plane graphs emerged from the study of triangulated terrains, which are polyhedral surfaces whose faces are triangles and whose intersection with any vertical line is at most one point. A set of guards (vertices or edges) monitors the surface of a terrain if every point on the terrain is visible from at least one guard in the set. The combinatorial aspects of the terrain guarding problems can be expressed as guarding problems on the plane triangulated graph underlying the terrain. Such graph is called a triangulation graph (triangulation, for short), because it is the graph of a triangulation of a set of points in the plane. These combinatorial aspects were studied in the 1990s [2, 3]. In this context of guarding plane graphs, where a set of guards needs to watch the faces of the graph, it was natural to extend the notions of monitoring by faces. So, three new concepts were defined by Hernández and Martins [4]: face-vertex covering, face guarding and face-edge covering in triangulations, that is, when faces monitor vertices, faces and edges, respectively. The monitoring concepts by vertices (vertex domination, vertex covering and vertex guarding) were extended to include its distance versions on plane graphs [5, 6], and the combinatorial bounds were established by Alvarado et al. [7]. The corresponding concepts by edges or faces at their distance versions (remote monitoring) were also introduced by Hernández and Martins [4] and they obtained combinatorial bounds for distance 2. In this paper, we generalize these results for any distance k 2 3 on a special class a triangulations, namely the maximal outerplanar graphs. In the next section, we first introduce some definitions and terminology used in this paper. In Section 3, we establish tight bounds for the minimum number of edges and faces that remote monitor the different elements of maximal outerplanar graphs. Finally, this paper concludes with Section 4 that discusses our results and future research.

Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

Abstract In this article we study some variants of the domination concept attending to the connectivity of the subgraph generated by the dominant set. This study is restricted to maximal outerplanar graphs. We establish tight combinatorial bounds for connected domination, semitotal domination, independent domination and weakly connected domination for any n-vertex maximal outerplaner graph.