Partitioning orthogonal polygons by extension of all edges incident to reflex vertices: lower and upper bounds on the number of pieces

Given an orthogonal polygon P, let |Π(P)| be the number of rectangles that result when we partition P by extending the edges incident to reflex vertices towards INT(P). In Tomás, A. P., Bajuelos, A. L., Marques, F.: Approximation algorithms to minimum vertex cover problems on polygons and terrains. In P.M.A Sloot et al. (Eds): Proc. of ICCS 2003, LNCS 2657, SpringerVerlag (2003) 869-878. we showed that |Π(P)| ≤ 1 + r + r 2, where r is the number of reflex vertices of P. We shall now give sharper bounds both for maxP |Π(P)| and minP |Π(P)|. Moreover, we characterize the structure of orthogonal polygons in general position for which these new bounds are exact.Programa de Financiamento Plurianual, Fundação para a Ciéncia e TecnologiaPrograma POSIPrograma POCTI, FCTFondo Europeo de Desarrollo Regiona

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