Santiago Canales

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.

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.

Combinatorial bounds on paired and multiple domination in triangulations

Abstract In this article we study some variants of domination in triangulation graphs. We establish tight combinatorial bounds for paired domination and 2-domination in maximal outerplanar graph, and for double domination and connected 2-domination in triangulations.

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.

Connecting Red Cells in a Bicolour Voronoi Diagram

Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted Voronoi diagram in O((n+m)2 log(nm)) time and for both the additively-weighted and power Voronoi diagram in O(nmlog(nm)) timeWe consider an extension of a question of Erdős on the number of k-gons in a set of n points in the plane. Relaxing the convexity restriction we obtain results on 5-gons and 5-holes (empty 5-gons). In particular, we show a direct relation between the number of non-convex 5-gons and the rectilinear crossing number, provide an improved lower bound for the number of convex 5-holes any point set must contain, and prove that the number of general 5-holes is asymptotically maximized for point sets in convex position.