Solving some Combinatorial Problems in grid n-ogons

Transcription

1 Solvng some Combnatoral Problems n grd n-ogons Antono L. Bajuelos, Santago Canales, Gregoro Hernández, Ana Mafalda Martns Abstract In ths paper we study some problems related to grd n-ogons. A grd n-ogon s a n-vertex orthogonal smple polygon, wth no collnear edges, that may be placed n a (n/)x(n/) square grd. We wll present some problems and results related to a subclass of grd n-ogons, the THIN grd n-ogons, n partcular a classfcaton for ths subclass of polygons. We follow by presentng the soluton of the MINIMUM VERTEX GUARD problem for the Mn-Area and for the Spral grd n-ogons. Fnally the soluton of the MAXIMUM HIDDEN VERTEX Set problem for THIN grd n-ogons s also presented. Keywords Art Gallery Problem, Grd n-ogon, Hdden Set, Orthogonal Polygon I I. INTRODUCTION N the feld of vsblty problems, guardng and hdng are among the most dstngushed and exhaustvely studed problems. In vsblty problems we are gven as nput a smple polygon (smple closed polygonal curve wth ts nteror). In guardng we need to fnd a mnmum number of guards postoned n the polygon, such that these guards collectvely see the whole polygon. Two ponts n the polygon see each other, f the lne segment connectng them les entrely n the polygon. In hdng, we need to fnd a maxmum number of postons n the polygon, such that no two of these postons see each other. The guardng problems started durng a conference, n 976, when Vctor Klee, posed the followng problem, whch today s known as the orgnal art gallery problem: How many statonary guards are needed to guard an art gallery room wth n walls? In the abstract verson of ths problem, the nput s a smple polygon P n the plane, representng the floor plan of the art gallery room and a guard s consdered a fxed Manuscrpt receved January 4, 007; Revsed receved May, 007 Ths work was supported n part by: the FCT (Portugal) fellowshp Grant SFRH/BD/865/006, the Grant MEC-HP and CAD-DPI (Span) and by CEOC (Portugal) trough Program POCTI, FCT, cofnanced by EC fund FEDER and by Acção Integrada Nº E-77/06. Antono L. Bajuelos, Assstant Professor of Computer Scence, Department of Mathematcs, Unversty of Avero & CEOC, Avero, Portugal (phone: (+35) , fax: ( , e-mal: lesle@ua.pt). Santago Canales, Assstant Professor of Computer Scence, Escuela Técnca Superor de Ingenera, Unvesdad Potfíca Comllas de Madrd, Span (e-mal: scanales@upcomllas.es). Gregoro Hernández, Assocate Professor of Computer Scence, Facultad de Informátca, Unversdad Poltécnca de Madrd, 8660 Madrd, Span (emal: gregoro@f.upm.es). Ana Mafalda Martns, Ph.D student of Computer Scence, Department of Mathematcs, Unversty of Avero & CEOC, Avero, Portugal (emal: mafalda.martns@ua.pt). Issue, Volume, pont n P wth π range vsblty. A set of guards covers P, f each pont of P s seen by at least one guard. Many varatons of the orgnal art gallery theorem have been studed over the years, such as: where the guards may be postoned (anywhere or n specfc postons, e.g., vertces), what knd of guards are to be used (e.g., statonary guards versus moble guards) and what assumptons are made for the nput polygon (such as beng orthogonal) (see [9]). The opposte problem of hdng a maxmum number of objects from each other n a gven smple polygon can have a practcal applcaton n computergames, where a player needs to fnd and collect or destroy as many objects as possble. Beng unable to see the next object whle collectng an object makes the game more nterestng. Such as the guardng problems, ths problem has many varatons []. In ths paper, of the guardng problems, we wll consder the MINIMUM VERTEX GUARD (MVG) problem that s the problem of fndng the mnmum number of guards placed on the vertces (vertex guards) needed to guard a gven smple polygon. And of the hdng problems, we wll consder the MAXIMUM HIDDEN VERTEX SET (MHVS) problem that s the problem of fndng the maxmum number of vertces of a gven smple polygon, such that no two vertces see each other. Both problems are NP-hard [3,7]. Important subclasses of polygons are the orthogonal smple polygons (smple polygons whose edges meet at rght angles). Indeed, they are useful as approxmatons to polygons; and they arse naturally n domans domnated by Cartesan coordnates, such as raster graphcs, VLSI desgn, or archtecture. The MVG and MHVS problems are stll NP-hard for orthogonal polygons. Ths paper has the ntenton of ntroducng a partcular type of orthogonal polygons - the grd n-ogons - that presents suffcently nterestng characterstcs that we are studyng and formalzng. Of the problems related to grd n-ogons, the vsblty problems are the ones that motvate us more, partcularly the guardng and hdng problems. The paper s structured as follows: n the next subsecton we wll ntroduce some prelmnary defntons and useful results. In secton 3 we wll present some problems and results related to THIN grd n-ogons, n partcular a classfcaton for these polygons. In secton 4 we wll expose some results related to the MVG problem on grd n-ogons and we wll study the MHVS problem on THIN grd n-ogons, a subclass of grd n-ogons. Fnally, n secton 5 we wll draw conclusons.

2 II. CONVENTIONS, DEFINITIONS AND SOME RESULTS In ths paper, the nteror and the boundary of a smple polygon P wll be denoted by INT(P) and BND(P), respectvely. And, for convenence, we wll assume that the vertces of P are ordered n a counterclockwse (CCW) drecton around INT(P). A vertex of P s called convex f the nteror angle between ts two ncdent edges s at most π, otherwse t s called reflex. We use r to represent the number of reflex vertces of P. It has been shown by O'Rourke that n= r+ 4, for every orthogonal smple polygon of n vertces (n-ogon, for short). A rectlnear cut of a n-ogon P s a partton of P obtaned by extendng each edge ncdent to a reflex vertex of P towards INT(P) untl t hts BND(P). We denote ths partton by Π(P) and the number of ts peces by Π(P). Each pece s a rectangle and so we call t a r-pece. A n-ogon that may be placed n a ( n / ) ( n / ) square grd and that does not have collnear edges s called grd n-ogon. We assume that the grd s defned by the horzontal lnes y =,, y/ and the vertcal lnes x =,, x/ and that ts northwest corner s (, ). Each grd n-ogon has exactly one edge n every lne of the grd. Grd n-ogons that are symmetrcally equvalent are grouped []. A grd n-ogon Q s called FAT ff Π(Q) Π(P), for all grd n-ogons P. Smlarly, a grd n-ogon Q s called THIN ff Π(Q) Π(P), for all grd n-ogons P. Let P be a grd n-ogon and r the number of ts reflex vertces. In [] t has been proven that, f P s FAT then 3r + 6r+ 4 Π ( P) =, for r even and 4 3( r + ) Π ( P) =, for r odd; f P s THIN then 4 ( P) = r+. There s a sngle FAT grd n-ogon (see Fg. (a)); however, THIN grd n-ogons are not unque (see Fg. (b)). (a) INTERNATIONAL JOURNAL of MATHEMATICS AND COMPUTERS IN SIMULATION Fg. : (a) The unque FAT grd n-ogons, for r =, 3 and 4; (b) Two THIN 0-ogons (b) boundary can be dvded nto a reflex chan and a convex chan. A polygonal chan s called reflex f ts vertces are all reflex (all except the vertces at the end of the chan) wth respect to the nteror of the polygon. And, a polygonal chan s called convex f ts vertces are all convex wth respect to the nteror of the polygon. In [6] t has been proven that there are SPIRAL grd n-ogons, for all n 6 ; however, they are not unque, as we may see ths n Fg. (b). And t was also proven that every SPIRAL grd n-ogon, wth r reflex vertces, s a THIN grd n-ogon. Gven a n-ogon P, we can assocate to Π(P) a graph, denomnated the dual graph of Π(P) and denoted by G(P), whch captures the adjacency relaton between peces of the partton. Each node of the dual graph corresponds to a pece of the partton and ts non-orented edges connect adjacent peces,.e., peces wth a common edge. We prove that f P s a THIN grd n-ogon then G(P) s a path graph,.e., a tree wth two nodes of vertex of degree, called leaves, and the remanng nodes of vertex of degree. To prove ths result we ntroduce Lemma.. (a) Fg. : (a) The unque MIN-AREA grd n-ogon, for r =, and 3; (b) Two dfferent SPIRAL grd -ogons (reflex chan s bold). Lemma.: Let P be a THIN (n+)-ogon. Then every grd n- ogon that yelds P by INFLATE-PASTE (a correct and complete method to generate grd n-ogons, well descrbed n [8]) s also THIN. Proposton.: Let P be a THIN grd n-ogon wth r (r ) reflex vertces. Then G(P) s a path graph (see examples n Fg. 3). The proof of ths proposton s done by nducton on r and uses lemma.. (b) The area of a grd n-ogon s the number of grd cells n ts nteror. In [] t has been proven that for all grd n-ogon P, wth n 8, r+ A( P) r + 3. A grd n-ogon P s a MAX-AREA grd n-ogon ff AP ( ) = r + 3 and t s a MIN- AREA grd n-ogon ff A( P) = r+. There are MAX-AREA grd n-ogons for all n, but they are not unque. However, there s a sngle MIN-AREA grd n-ogon and ts form s llustrated n Fg. (a). Regardng MIN-AREA grd n-ogons, t s obvous that they are THIN grd n-ogons, because Π ( P) = r + holds only for THIN grd n-ogons. However, ths condton s not suffcent for a THIN grd n-ogon to be a MIN-AREA grd n-ogon. A grd n-ogon s called a SPIRAL grd n-ogon f ts Fg. 3: Three THIN grd 0-ogons and respectve dual graphs. Proposton.3: Let P be a grd n-ogon, wth n > 6. If P s not THIN then G(P) s not a tree (see example n Fg. 4). Fg. 4: A grd 0-ogon and respectve dual graph. Issue, Volume,

3 Proposton.. establshes that, beng P a THIN grd n- ogon, G(P) s a path graph. So, each r-pece of Π(P) s adjacent to at most two r-peces. In ths way, each r-pece has at most nteror edges. Consequently, we conclude that n Π(P) there are 3 types of r-peces: Type : wth one nteror edge and three boundary edges; Type : wth two nteror edges not adjacent and two boundary edges not adjacent; Type 3: wth two adjacent nteror edges and two adjacent boundary edges. The r-peces of the Type correspond to leaves of G(P) and those of the Type and Type 3 correspond to nodes of degree. We showed that of the 4 vertces of the r-peces of Type three are vertces of P, beng two reflex and the other convex, and the other vertex s an nteror pont of an edge of P. Of the 4 vertces of the r-peces of Type two are convex vertces of P and the other two are nteror ponts of edges of P. And fnally, of the 4 vertces of the r-peces of Type two are vertces of P, beng one reflex and the other convex, and the other two are nteror ponts of edges of P (see Fg. 5). Fg. 6: THIN wth r = 4; on the left s represented ts dual graph and on the rght ts skeleton. Lemma.4: The skeleton of a THIN grd n-ogon s an orthogonal polygonal curve wth r+ vertces. III. SOME PROBLEMS RELATED TO THIN GRID N-OGONS As we have seen n Secton, on the contrary of the FATs, the THIN n-ogons are not unque. In fact, there are THIN 8- ogon, 30 THIN 0-ogons, 49 THIN -ogons, etc. Thus, t s nterestng to evdence that the number of THIN grd n-ogons ( THIN(n) ) grows exponentally. Does there exst some expresson that relates n to THIN(n)? As a step for the resoluton of ths problem we wll frst group the THINs nto classes. In Secton we defned the skeleton of a THIN grd n- ogon. Now, usng ths concept, we wll group THIN grd n- ogons nto classes. From the skeleton of the THIN grd n-ogon, we can always represent t by a chan of 0's and 's, wth length r. For that we proceed n the followng way: we transverse ts skeleton, startng at vertex u, and then we represent each turn left by and each turn rght by 0. Now, we wll defne two operatons on these chans: the complementary operaton and the nverson operaton. Type Type Type 3 Fg. 5: Types of r-peces of a THIN grd n-ogon. Now, we are gong to defne the skeleton of a THIN grd n- ogon. Let P be a THIN grd n-ogon. Snce G(P) s a path graph, we can say that P has two extremes : the r-peces assocated wth the leaves of the dual graph. We wll denote by kernel the extreme that has the horzontal edge wth hghest y-coordnate. From ths graph we can obtan an orthogonal polygonal curve (.e., a polygonal curve wth horzontal or vertcal edges) n the followng way: we take the centrod of each r-pece, then we connect each one wth the centrods of the adjacent r-peces and, fnally, we remove the central vertex of each three algned vertces, as we can see n Fg. 6. We choose, for the frst vertex of ths orthogonal curve the kernel's centrod. Therefore t s easy to prove the next result. Defnton 3.: Let c be a chan of 0's and 's, wth length r,.e., c = b b b r, where b = 0 or b =, for =,,, r. The complementary operaton s an operaton whch takes c as the argument and returns ts complementary c* = b *b * b r *, where b *= f b = 0 and b * = 0 f b =, =,,, r. The nverson operaton s an operaton whch takes c as the argument and returns ts nverse c - =b r b r- b. For example, the complementary of the chan c = 000 s the chan c* = 000 and ts nverse s c - = 000. Easly we can verfy that, (c*) - = (c - )*, (c*)* = c and (c - ) - = c. Proposton 3.: Let C r be the set of all chans, of 0's and 's, wth length r. The relaton ~ defned on C r by c ~ c c = c c = c - c = c * c = (c * ) -, s an equvalence relaton. Consder, now, the quotent set of C r by ~, C r / ~ = {[c ] ~ : c C r }. Note that, each equvalence class has more than one representatve. We assume that the representatve of each equvalence class always starts by. Proposton 3.3: Let P r be the set of all THIN grd n-ogons, wth r reflex vertces. The relaton defned on P r by P P c ~ c, where c and c are the chans that represent P and P, respectvely, s an equvalence relaton. The proof of ths proposton s trval. Let P r / = {[P ] : P C r }. Let P, P P r and c, c C r the chans that represent them, respectvely. Note that, P and P belong to the same class (.e., P and P are equvalents) f one of the followng condtons s true: () c = c ; () c = c - ; () c = c * or (v) c = (c *) -. Observe that, geometrcally, () can correspond to a horzontal reflecton and () to a vertcal reflecton. In Fg. Issue, Volume,

4 7 sx THINs wth 4 reflex vertces that belong to the same class are llustrated. the begnnng of the polygon, t s ndfferent the one that s modfed. Nevertheless, when the polygon has two collnear edges, the choce of the edge s not always ndfferent. Step Fg. 7: THINs wth 4 reflex vertces and respectve chans. We can place the followng queston: Let c be chan of 0's and 's wth length r, started by. Is t always possble to construct a THIN, wth r reflex vertces, whose chan that represents t s c? To answer ths queston we present the next algorthm: Step Step 3 Algorthm 3.4: Construct a THIN form a chan of 0's and 's, of length r Let c be a chan of 0's and 's, of length r, started by.. From the chan draw a skeleton gnorng collneartes.. Move e vertcal sweep lne from left to rght to elmnate vertcal collneartes. Repeat the prevous step untl there are no more collnear vertcal edges. 3. Move a horzontal sweep lne from bottom to top to elmnate horzontal collneartes. Repeat the prevous step untl there are no more collnear horzontal edges. Fgure 8 llustrate ths algorthm from the chan 0. Fg. 9: Constructng the THIN from the chan 0 Anyhow, ths algorthm always generates a THIN grd n- ogon whose chan that represents t s equvalent to c. Thus, f the chan that represents the Thn, generated by ths algorthm, s c*; c - or (c*) -, t s enough to make a vertcal reflecton, a horzontal reflecton or a vertcal reflecton followed by a horzontal reflecton, respectvely, so that the chan that represents t s exactly c, see Fg.0 for llustraton. Thus, ths algorthm proves that for each chan of 0's and 's, wth length r, started by, there s a Thn grd n-ogon wth r reflex vertces represented by t. Step Step Step 3 Fg. 8: Constructng the THIN for the chan 0 Important remarks:. In the Step, f we construct a skeleton, gnorng the collneartes, we can obtan a skeleton that not correspond to the gven chan, for example n the Fg. 9, the chan that represents the constructed Thn s (complementary followed by nverson of the gven chan).. To elmnate collneartes, n step and 3, t s necessary to modfy the edge correspondng to the begnnng of the polygon. If two edges correspond to the begnnng of the polygon, or no edge correspond to Fg. 0: (a) The chan that represents the THIN after the reflectons s c = 0; (b) The chan that represents the THIN after the reflectons s c = 00. Based on the prevous reasonng and defntons t s not dffcult to prove the followng result. Proposton 3.5: The correspondence f: P r / C r / ~ defned by f([p ]) = [c ], where c C r s the chan that represents P P r, whch s a representatve of the class [P ], s a bjectve functon. Issue, Volume,

5 Now, we may show the next result that allows us to count the number of classes of THIN grd n-ogons wth r reflex vertces. Proposton 3.6: The number of classes of THIN grd n-ogons wth r reflex vertces (r ) s equal to: ( r 3) r +, f r s odd ( r ) r +, f r s even Proof. By proposton 3.5 we can conclude that P r / = C r / ~, so we just have to calculate C r / ~. The cardnal of C r s r and the number of symmetrcal chans (c = c - ), wth length r, s / r. If a chan c s symmetrcal, then ts equvalence class s consttuted by two chans, c and c*. If a chan c s not symmetrcal, to fnd the cardnal of ts class, we have to dstngush two cases: r odd and r even. If r s odd, all the chans have 4 equvalent chans: c, c -, c* and (c*) - (for example: c = 00, 00, 000 and 000). If r s even, there are chans that have 4 equvalent chans (e.g., c = 0) and chans that only have equvalent chans; ths case happens when c* = c - (e.g., for the chan c = 00, c* = c - = 00. Let us now count the number of equvalence classes. If r s odd, the number of equvalence classes of the symmetrcal r+ r chans (SC) s SC = = and the number of equvalence classes of the non symmetrcal chans (NSC) s r+ r 3 r r NSC = =. Thus, f r s odd, the total 4 4 number of equvalence classes s ( 3 r ) r r 3 r + = + r. If r s even, the number of equvalence classes of r ( ) symmetrcal chans s equal to r SC = =. The number of equvalence classes of non symmetrcal chans consttuted by two chans (for example, the classes of the r ( ) chans 000, 00, 000, ) s r =. In fact, to obtan c* = c -, the second half of the chan s completely determned by the frst half. Therefore, the cardnal of these classes s half of the number of chans of ths type. And, the number of equvalence classes of non symmetrcal chans consttuted by four chans s r r ( * All Symmetrc Chans wth c = c ) = r = 4 4 ( r ) r. Thus, f r s even, n the total, the number of ( ) r equvalence classes s equal to + r INTERNATIONAL JOURNAL of MATHEMATICS AND COMPUTERS IN SIMULATION However, there are stll some open problems to solve, such as: How many elements THIN grd n-ogons does each class have? Wll t be possble to fnd an algorthm that generates all THIN grd n-ogons of the same class? Note that, solvng the frst problem we also solve the ntal problem, that s: Does there exst some expresson that relates n to THIN(n)? IV. SOME VISIBILITY PROBLEMS ON GRID n-ogons Of the problems related to grd n-ogons, the guardng and hdng problems are the ones that motvate us more, partcularly the MVG and MHVS problems. Snce THIN and FAT n-ogons are the classes for whch the number of r-peces s mnmum and maxmum, we thnk that they can be representatve of extremal behavor. Besdes that they are used expermentally to evaluate approxmate methods of resoluton of the MVG problem, so we started wth them. We have already proven that to guard any FAT grd n-ogon t s always suffcent two π/ vertex guards (vertex guards wth π/ range vsblty) and establshed where they must be placed [4]. However, THIN grd n-ogons are much more dffcult to guard, n spte of havng much fewer r-peces than FATs. Besdes, they are not unque, so we tred to characterze structural propertes of classes of THINs that allow for smplfyng the problem's study. Up to now the only qute characterzed subclasses are the MIN-AREA and the SPIRAL grd n-ogons. We proved that to guard any MIN-AREA and SPIRAL grd n- ogon n/6 and n/4 vertex guards are necessary, respectvely. Moreover, we showed where those guards could be placed [5, 6]. MAXIMUM HIDDEN VERTEX Set Problem on THIn grd n-ogons Gven a smple polygon, P, and a subset of vertces of P, HV, we say that HV s a hdden vertex set f no two vertces n HV see each other. The MAXIMUM HIDDEN VERTEX SET problem on a smple polygon asks for an hdden vertex set, HV, of maxmum cardnalty. We wll call the elements of HV hdden vertces. Shermer [7] proved that the sze of the MAXIMUM HIDDEN VERTEX SET of a n-ogon s at most (n-)/. Ths tght bound s acheved n starcase polygons. We wll show that, gven a THIN grd n-ogon the maxmum cardnalty of a hdden vertex set s n/4. Let P be a THIN grd n-ogon and S = u u u n/ ts skeleton. Let us assume, wthout loss of generalty that the frst edge of S, [u u ], s horzontal and that u s to the rght of u. Note that, the boundary of P conssts of two joned polygonal chans, c and c, parallel to S, where the frst edge of c s a bottom edge and the frst edge of c s a top edge. Note that, c and c can be expressed as ordered sequences of vertces c v v... v n = and / c v v... v n =, where / v denotes the th vertex of c and v denotes the th vertex of c (see Fg. ). Ths way, BND(P) = c vn/ v n/ c vv. Observe, also, that, f we transverse S, startng at vertex u, c s always on the rght of S and c on the left. Issue, Volume, 007 8

6 Fg. : Two THINs grd n-ogons, ts skeletons and the chans c and c (c n bold). In Case the vertex that s marked as hdden s the vertex vk and n Case s the vertex v k. In both cases the marked vertex does not see none of the already marked as hdden, snce of the already vsted vertces ths one only sees v and v (see Fg.4). k k To each vertex of the skeleton we correspond two vertces of the polygon, one n c and another one n c. That s, to u S, we correspond the vertces v c and v c. And to each edge of the skeleton we correspond two parallel edges of the polygon, one n c and another one n c. That s, to uu + S, we correspond the edges c and c. Note that, by constructon of the skeleton, we can easly see that any pont of sees any pont of. Now, for each u k- S wth k =,, n/4, we mark an hdden vertex n P, n the followng way: for k = we mark v ; for k we mark v or v, dependng f v s k k k reflex or convex, respectvely (see Fg., for llustraton). Fg. 4: The shaded zones are not vsble by the marked vertces. Observe that f we mark as hdden the vertces v k (n Case ) and v k (n Case ) we can't guarantee that they don't see any of the vertces already marked as hdden, snce they see more backwards (see Fg. 5). Fg. 5: The shaded zones are not vsble by the marked vertces. Fg. : Two THINs n-ogons and marked hdden vertces (c n bold). Note that, the n/4 marked vertces form an hdden vertex set, snce each tme that we mark a new vertex as hdden we can guarantee that t does not see none of the vertces that prevously had been marked as hdden. In fact, for k = t s trval. For k, we have two cases, dependng f v k s reflex (Case ) or convex (Case ), as we can se n Fg.3. Fg. 3: In Case the vertex v v k k s reflex and n Case the vertex s convex. Therefore t s easy to prove the next result. Lemma 4.: For any THIN grd n-ogon there s an hdden vertex HV and HV = n/4. And then usng ths Lemma we prove our man result. Theorem 4.: Let P be a THIN grd n-ogon. The maxmum cardnalty of an hdden vertex set n P s n/4. V. CONCLUSION We presented some results related to grd n-ogons. Of the hdng problems related to the grd n-ogons, t s the MHVS problem that motvates us more. We proved that the maxmum cardnalty of an hdden vertex set n a THIN n-ogon s n/4. Moreover, we establshed a possble postonng for those hdden vertces. We also establshed a possble classfcaton for THIN n-ogon, as a step to launch an expresson that relates n to THIN(n). Issue, Volume, 007 8

7 ACKNOWLEDGMENT We wsh to thank Inês Perera Matos from the Unversty of Avero for helpful dscussons about the problem of classfcaton of THIN grd n-ogons. REFERENCES [] A. L. Bajuelos, A. P. Tomás: Parttonng Orthogonal Polygons by Extenson of All Edges Incdent to Reflex Vertces: lower and upper bounds on the number of peces. Proc. of ICCSA 004, LNCS 3045, Sprnger-Verlag (004) [] S. Edenbenz: In-approxmablty of Fndng Maxmum Hdden Sets on Polygons and Terrans. Computatonal Geometry (00) [3] D. Lee, A. Ln: Computatonal Complexty of Art Gallery Problems. IEEE Transactons on Informaton Theory IT-3 (986) [4] A. M. Martns, A. Bajuelos: Some Propertes of Fat and Thn grd n- ogons. n Wley-VCH Verlag (Eds): Proc. of ICNAAM 005 (005) [5] A. M. Martns, A. Bajuelos: Characterzng and Coverng some Classes of Orthogonal polygons. In Proc. ICCS 006, LNCS 399, Sprnger- Verlag (006) [6] A. M. Martns, A. Bajuelos: Vertex Guards n a Subclass of Orthogonal Polygons. Internatonal Journal of Computer Scence and Network Securty (IJCSNS), Vol. 6 No. 9, September 006. [7] T. Shermer: Hdng people n polygons, Computng 4 (989) [8] A. P. Tomás, A. Bajuelos: Quadratc-Tme Lnear-Space Algorthms for Generatng Orthogonal Polygons wth a Gven Number of Vertces. In Proc. of ICCSA 004, LNCS 3045, Sprnger-Verlag (004) 7-6. [9] J. Urruta: Art Gallery and Illumnaton Problems. In J.-R. Sack and J. Urruta (edtors), Handbook of Computatonal Geometry, Elsever (000). Issue, Volume,