Finite Order Domination in Graphs

Transcription

1 Fiite Order Domiatio i Graphs AP Burger, EJ Cockaye, WR Grüdligh, CM Myhardt, JH va Vuure & W Witerbach September 11, 003 Abstract The (previously studied) otios of secure domiatio ad of weak Roma domiatio ivolve the costructio of protectio strategies i a simple graph G = (V, E), by utilisig the miimum umber of guards eeded at vertices i V to protect G i differet scearios (these miimum umbers are called the secure [weak Roma] domiatio parameters for the graph). I this paper these otios are geeralised i the sese that safe cofiguratios i G are ot merely sought after oe move, but rather after each of k 1 moves. Some geeral properties of these geeralised domiatio parameters are established, after which the parameter values are foud for certai simple graph structures (such as paths, cycles, multipartite graphs ad products of complete graphs, cycles ad paths). Keywords: Secure & weak Roma Domiatio, Higher Order Domiatio, Graph Protectio. AMS Subject Classificatio: 05C69. 1 Itroductio A guard fuctio for a graph G = (V, E) is a mappig f : V {0, 1,,...} such that f(v) deotes the umber of guards statioed at a vertex v V. A guard fuctio partitios the vertex set of G ito subsets V i = {v : f(v) = i}, i = 0, 1,,... ad we (imprecisely) write f = (V 0, V 1, V,...). A guard Departmet of Applied Mathematics, Stellebosch Uiversity, Private Bag X1, Matielad, 760, Republic of South Africa, fax: , vuure@su.ac.za. Departmet of Mathematics ad Statistics, Uiversity of Victoria, Box 3045, Victoria, BC, Caada, V8W 3P4, cockaye@math.uvic.ca 1

2 fuctio is called safe if each v V 0 is adjacet to some u V \V 0 (i.e. if V \V 0 is a domiatig set of G). The weight of a guard fuctio is deoted w(f) = v V f(v). The followig four kids of safe guard fuctios have bee studied i the literature: 1. A domiatig fuctio (DF) is a safe guard fuctio f = (V 0, V 1 ). Note that f = (V 0, V 1 ) is a DF if ad oly if V 1 is a domiatig set of G. The miimum weight of a DF is the well kow lower domiatio umber, γ(g) = mi DFs V 1. See, for example, [5] for kow results o this parameter.. Prompted by a strategy employed by Roma emperor Costatie, first Stewart [7] ad the Cockaye, et al. [1], exteded the otio of a DF to iclude so called Roma domiatio. A Roma domiatig fuctio (RDF) is a safe guard fuctio f = (V 0, V 1, V ) such that each v V 0 is adjacet to some u V. The miimum weight of a RDF is deoted γ R (G) = mi RDFs ( V 1 + V ), which is called the Roma domiatio umber of G. 3. As a result of usig possibly too may guards i a RDF, Heig & Hedetiemi [6] suggested relaxig the defiitio somewhat to arrive at the otio of so called weak Roma domiatio. A weak Roma domiatig fuctio (WRDF) is a safe guard fuctio f = (V 0, V 1, V ) with the property that each v V 0 is adjacet to some u V 1 V such that 1, if s = v g(s) = f(u) 1, if s = u f(s), if s V \{u, v} is also a safe guard fuctio. I this case we write g = move(f, u v) to mea that g is the safe guard fuctio obtaied from the safe guard fuctio f by movig a guard from vertex u to a adjacet vertex v, ad leavig all other guards uchaged. The miimum weight of a WRDF is deoted γ r (G) = mi ( V 1 + V ), WRDFs which is called the weak Roma domiatio umber of G.

3 3 4. Fially, the defiitio of weak Roma domiatio was broadeed yet further by Cockaye, et al. [3] to iclude the otio of secure domiatio. A secure domiatig fuctio (SDF) is a safe guard fuctio f = (V 0, V 1 ) with the property that each v V 0 is adjacet to some u V 1 such that g(s) = move(f, u v) 1, if s = v = 0, if s = u f(s), if s V \{u, v} is also a safe guard fuctio. The miimum weight of a SDF is deoted γ s (G) = mi V 1, SDFs which is called the secure domiatio umber of G. I [1, 3, 6] it was show that, for ay coected graph G, γ(g) γ r (G) { γr (G) γ(g) γ s (G). (1) A umber of iterestig properties of these four parameters have bee established i [1, 3, 6], ad these parameters have also bee determied for simple graph classes, such as complete graphs, paths, cycles ad complete multipartite graphs. I [, 3] bouds for these parameters are established for more complex graph structures, such as grid graphs, products of cycles, products of complete graphs ad claw free graphs. Fially, the geeral lower boud γ s (G) ( t + 5) ( + 1) (t 1)(t ) was proved i [] for ay K t free graph G of order ad maximum degree. Foolproof geeralisatios There is a fudametal differece betwee domiatig fuctios ad Roma domiatig fuctios o the oe had, ad weak Roma domiatig fuctios ad secure domiatig fuctios o the other: the first two ivolve static cofiguratios of guards o the vertices of G, while the secod two ivolve movig a guard from oe vertex to a adjecet vertex ad are therefore dyamic. We shall oly cosider dyamic cofiguratios of ()

4 4 guards. The defiitios of both weak roma domiatio ad secure domiatio are smart i the followig sese: they require that, for ay uoccupied vertex v experiecig a problem, there is a guard at a adjacet vertex u such that movig the guard from u to v results i a safe guard fuctio. It is the strategist s task to determie which move to make i order to resolve the problem at vertex v. It is therefore possible to defie foolproof versios of these dyamic cofiguratio cases (where a strategist is ot required): (5) A foolproof weak Roma domiatig fuctio (FWRDF) is a safe guard fuctio f = (V 0, V 1, V ) such that, for each u V 1 V i the (ope) eighbourhood of ay v V 0, the fuctio g(s) = move(f, u v) 1, if s = v = f(u) 1, if s = u f(s), if s V \{u, v} is also a safe guard fuctio. The miimum weight of a FWRDF is deoted γr (G) = mi ( V 1 + V ), FWRDFs which is called the foolproof weak Roma domiatio umber of G. (6) A foolproof secure domiatig fuctio (FSDF) is a safe guard fuctio f = (V 0, V 1 ) such that, for each u V 1 i the (ope) eighbourhood of ay v V 0, the fuctio g(s) = move(f, u v) 1, if s = v = 0, if s = u f(s), if s V \{u, v} is also a safe guard fuctio. The miimum weight of a FSDF is deoted γ s (G) = mi FSDFs V 1, which is called the foolproof secure domiatio umber of G. Whe referrig to the previously studied versios of these defiitios (as opposed to the ew defiitio versios give above), we shall use the term smart secure [weak Roma] domiatio istead of foolproof secure [weak Roma] domiatio i order to distiguish betwee the two kids of dyamic cofiguratios. Note that, for ay graph G, γ r (G) γ r (G) ad γ s (G) γ s (G), (3)

5 5 sice foolproof secure [weak Roma] domiatio requires a more robust cofiguratio tha does its smart couterpart. We also have the followig (less trivial) result. Propositio 1 For ay graph G, γ r (G) γ R (G). Proof: Suppose f R = (V 0, V 1, V ) is a RDF for the graph G. The f R is certaily a safe guard fuctio for G, ad ay move of the form g r = move(f R, u v) clearly results i a safe guard fuctio g r i the case where u V. Furthermore, if u V 1, the g r is also a safe guard fuctio, sice i this case the sole possible reaso for icludig u i V 1 i the first place, was that u should domiate itself. But after the move u is domiated by v. The followig geeralisatio of the bouds i (1) is therefore possible, by utilisatio of (3) ad Propositio 1. Corollary 1 For ay graph G, γ(g) γ r (G) { γ r (G) γ R (G) γ(g) γ s (G) γ s (G). (4) It is, of course, possible to establish values for the dyamic cofiguratio parameters i the foolproof case for simple graph classes, as has bee doe for the smart cases i [1, 3, 6]. We demostrate this for the case of foolproof secure domiatio of paths. Theorem 1 For ay path P, γ s (P ) =. Proof: It is first show, by cotradictio, that γs (P ). Suppose f = (V 0, V 1 ) is a FSDF for the path P : v 1 v with V 1 <. The at least oe of the followig situatios occur (without loss of geerality, by choice of the vertex labellig): (i) Ú ½ Ú ¾ Ú (ii) Ú Ú ½ Ú ¾ Ú Ú where dark vertices deote elemets of V 1. Suppose (i) holds. The g = move(f, v v 3 ) is ot a safe guard fuctio, sice v 1 V 0 is ot adjacet to ay u V 1. Now suppose (ii) holds. The g = move(f, v i+3 v i+4 ) is ot a safe guard fuctio, sice v i+ V 0 is ot adjacet to ay u V 1. These cotradictios show that γs (P ). (5)

6 6 To see that γ s (P ), (6) observe that f odd = (Ṽ0, Ṽ1), with Ṽ1 = {v i : i 1 (mod )} ad Ṽ0 = V (P )\Ṽ1 is a FSDF for P if is odd. Otherwise f eve = ( V 0, V 1 ) with V 1 = {v i : i, 3 (mod 4)} ad V 0 = V (P )\ V 1 is a FSDF for P if is eve. The desired result follows by a combiatio of (5) ad (6). Similar results to the oe above for other graph classes or for the parameter γ r are possible at this poit, but we refrai from provig such results util we have established a more geeral settig for smart ad foolproof domiatio. 3 Higher order geeralisatios It is atural to geeralise the otios of smart [foolproof] secure ad weak Roma domiatio so that safe cofiguratios are guarateed after each of k 1 moves to a sequece of problem vertices (heceforth iformally called a problem sequece) istead of cosiderig oly oe problem vertex i the graph at a time. The followig four defiitios achieve just such a geeralisatio. (7) A smart k weak Roma domiatig fuctio (k SWRDF) is a safe guard fuctio f (0) = (V (0) 1, V (0) ) with the property that, for ay sequece of vertices v 0, v 1,... v k 1, there exists a sequece of vertices u i V (i) 1 V (i) i the eighbourhoods of v i such that the fuctios f (i+1) (s) = move(f (i), u i v i ) are also safe guard fuctios for all i = 0,..., k 1. The miimum weight of a k SWRDF is deoted γ r,k (G) = mi k SWRDFs ( V (0) 1 + V (0) which is called the smart k weak Roma domiatio umber of G. (8) Similarly, a foolproof k weak Roma domiatig fuctio (k FWRDF) is a safe guard fuctio f (0) = (V (0) 1, V (0) ) with the property that, for ay sequece of vertices v 0, v 1,..., v k 1, the fuctios f (i+1) (s) = move(f (i), u i v i ) are also safe guard fuctios for ay sequece of vertices u i V (i) 1 V (i) i the eighbourhoods of v i ad all i = 0,..., k 1. The miimum weight of a k FWRDF is deoted ( ) γr,k(g) = V (0) 1 + V (0), mi k FWRDFs which is called the foolproof k weak Roma domiatio umber of G. ),

7 7 (9) A smart k secure domiatig fuctio (k SSDF) is a safe guard fuctio f (0) = (V (0) 1 ) with the property that, for ay sequece of vertices v 0, v 1,..., v k 1, there exists a sequece of vertices u i V (i) 1 such that the fuctios f (i+1) (s) = move(f (i), u i v i ) are also safe guard fuctios for all i = 0,..., k 1. The miimum weight of a k SSDF is deoted γ s,k (G) = (0) mi V 1, k SSDFs which is called the smart k secure domiatio umber of G. (10) Similarly, a foolproof k secure domiatig fuctio (k FSDF) is a safe guard fuctio f (0) = (V (0) 1 ) with the property that, for ay sequece of vertices v i (i = 0,..., k 1), the fuctios f (i+1) (s) = move(f (i), u i v i ) are also safe guard fuctios for ay sequece of vertices u i V (i) 1 i the eighbourhoods of v i ad all i = 0,..., k 1. The miimum weight of a k SSDF is deoted γ s,k(g) = (0) mi V 1, k FSDFs which is called the foolproof k secure domiatio umber of G. We therefore have the special cases γ r = γ r,1, γ r = γ r,1, γ s = γ s,1 ad γ s = γ s,1. The case k = 0 correspods to the situatio where o moves are allowed. So, for coveiece, let γ r,0 = γ r,0 = γ s,0 = γ s,0 = γ. Furthermore, the followig relatioships betwee the smart ad foolproof versios of the ewly defied parameters trivially hold, as a geeralisatio of (3). Propositio For ay graph G ad ay k N, γ r,k (G) γr,k (G) ad γ s,k (G) γs,k (G). The followig growth relatioships of the parameters with respect to icreasig values of k hold. Propositio 3 For ay graph G ad ay k N 0, (a) γ r,k (G) γ r,k+1 (G), (b) γ r,k (G) γ r,k+1 (G), (c) γ s,k (G) γ s,k+1 (G), (d) γ s,k (G) γ s,k+1 (G).

8 8 Proof: (a) Ay (k + 1) WRDF with miimum weight γ r,k+1 (G) for G is also a k WRDF for G, ad the weight of this last domiatig fuctio is bouded from below by γ r,k (G). The proofs of parts (b) (d) are similar. It is easy to see that the followig result is true. Lemma 1 For ay graph G ad ay edge e E(G), γ r,k (G) γ r,k (G e) ad γ s,k (G) γ s,k (G e), for all k N 0. This lemma may be used repeatedly to prove the followig two results. Propositio 4 If the vertex set of a graph G is partitioed ito two subsets S 1 ad S, the, for all k N 0, (a) γ r,k (G) γ r,k ( S 1 ) + γ r,k ( S ), (b) γ s,k (G) γ s,k ( S 1 ) + γ s,k ( S ). Propositio 5 If H is a spaig subgraph of G, the γ r,k (G) γ r,k (H) ad γ s,k (G) γ s,k (H) for all k N 0. Note that, i geeral, it is ot possible to establish results similar to those of Lemma 1 ad Propositios 4 ad 5 for the parameters γr,k (G) ad γs,k (G). For example, removig a edge may icrease or decrease the value of γs,1. This may be see by observig that γs,1(p 4 ) =, while γs,1(p 4 e) = γs,1(p 1 ) + γs,1(p 3 ) = 3. O the other had γs,(p 4 ) = 3, yet γs,(p 4 e) = γs,(p ) + γs,(p ) =. It is easily see that the followig result is true. Propositio 6 If G is a order graph such that, for some subset of vertices S = {v 1,..., v m } V (G), the graph G S possesses a perfect matchig, the, for all k N 0, γ s,k (G) m + m = m +. Fially, we coclude this sectio with a summary of relatioships betwee the various parameters cosidered i this sectio, as a geeralisatio of (4). Theorem The relatioships γ(g) γ r,k (G) γ s,k (G) γ(g) γ r,k (G) γ s,k (G) (7) hold for all k N.

9 9 4 Parameters for special graphs I this sectio we cosider a umber of simple graph classes ad fid values for the four ew fiite order domiatio parameters cosidered i this paper. 4.1 Paths For paths we establish three of the four fiite order parameter values precicely, as summarised i the followig theorem. Theorem 3 For ay path P,, k + 1 (a) γ r,k (P ) = γ s,k (P ) = 4k + 3, for all k N 0, { (b) γs,k (P k+1 ) = k+3 if k 1 if k 1. Proof: (a) It is first show that γ s,k (P ) k+1 4k+3 for the path P : v 1 v. Partitio the path P ito (l) 4k+3 subpaths P 4k+3 of legth 4k+3 (l = 1,..., 4k+3 ) ad oe (possibly empty) subpath P c : u 1 u u c of legth c (mod 4k+3), ad cosider the fuctio f = (V (0) 1 ), where V (0) 1 = {v i : i (mod 4k + 3) 0 (mod ), v i V (P4k+3 l )} {u j V (P c ) : j 1 (mod )} ad V (0) 0 = V (P )\V (0) 1. We shall show that f is a k SSDF for P, by showig that for ay sequece of k problem vertices there exist m k moves that reders safe guard fuctios f (i) for P (i = 0,..., k). È ½µ Þ Ð ß Þ Ð ß Þ Ð ß È Ò µ Ú½ Ú¾ Ú Ù½ Ù¾ Ù È We oly have to cosider the case where the whole sequece of problem vertices occurs i oe subpath P (l) 4k+3, because if there exists a move sequece that reders safe guard fuctios f (i) for P (l) 4k+3 (i = 0,..., k), give ay sequece of k problem vertices withi the subpath, the there also exists a move sequece that reders safe cofiguratios for fewer problem vertices withi the subpath. Sice either P c or P c v possesses a perfect matchig, it follows by Propositio 6 that γ s,k (P c ) = c/ ad hece there exists a move sequece wholly withi P c that reders safe cofiguratios for P c, give ay sequece of k 1 problem vertices i P c. Therefore cosider, without loss of geerality, a sequece of k problem vertices v ij V (P (1) 4k+3 ), j = 1,..., k. We cosider two mai cases:

10 10 Case A: v ij {v 4k+1, v 4k+, v 4k+3 } for all j = 1,..., k. I this case M 1 = k i=1 v i 1, v i is a perfect matchig of the subpath v 1,..., v 4k. Therefore γ s,k (P (1) 4k+3 ) k + 1, by utilisatio of Propositios 6 ad 4(b) ad Theorem. Case B: v ij {v 4k+1, v 4k+, v 4k+3 } for some j {1,..., k}. I this case we distiguish betwee two further subcases: Subcase B(i): v ij {v 1, v, v 3 } for all j = 1,..., k. I this subcase M = v i, v i+1 k+1 i= is a perfect matchig of the subpath v 4,..., v 4k+3. Hece we have, by a similar argumet as i Case A, that γ s,k (P (1) 4k+3 ) k + 1. Subcase B(ii): v ij {v 1, v, v 3 } for some j {1,..., k}. I this subcase there are at most k problem vertices i the subpath v 4,..., v 4k. But the it follows, by the pigeohole priciple, that there are at least 4 cosecutively labelled vertices that are ot problem vertices: suppose they are v l, v l+1, v l+, v l+3 (the case where the first of these labels is odd, is similar). The M 3 = l v i 1, v i ad M 4 = i=1 k+1 i=l+ v i, v i+1 are perfect matchigs of the subpaths P := v 1,..., v l ad P := v l+4,..., v 4k+3 respectively, ad we have, agai by Theorem ad Propostios 6 ad 4(b), that γ s,k ( P P ) k. Hece γ s,k (P (1) 4k+3 ) k + 1, because γ( v l+1, v l+, v l+3 ) = 1. Cosequetly we have, i all cases, that c k + 1 (γ r,k (P ) ) γ s,k (P ) (k + 1) + 4k + 3 4k + 3. (8) (The last iequality ca be proved by first showig that c = k+1 4k+3c if c < 4k + 3.) To prove that k + 1 (γ s,k (P ) ) γ r,k (P ) 4k + 3, (9)

11 11 suppose, to the cotrary, that γ r,k (P ) (k+1) 4k+3 1 = 4k+3 (k + 1) + c (i) 1. The there will be a subpath P 4k+3, i {1,..., 4k+3 }, cotaiig at most k vertices from V (0) 1 V (0), or else P c will cotai at most c 1 vertices from V (0) 1 V (0). We cosider the former possibility first. Suppose, without loss of geerality, that P (1) 4k+3 cotais at most k vertices from V (0) 1 V (0). Cosider the set of problem vertices I = {v 4l 1 : l = 1,..., k}. Because this is a idepedet set, I V (k) 1 V (k). Furthermore, because f (k) must be a safe guard fuctio, J = {v 4l+1 : l = 0,..., k} must be domiated by vertices i V (k) 1 V (k). But o vertex i I is adjacet to vertices i J. Therefore (V (k) 1 V (0) ) V (P (1) 4k+3 ) k + 1, wh ich is a cotradictio. Fially, if P c cotais oly c (0) 1 vertices from V 1 V (0), the we get a similar cotradictio by cosiderig the problem vertex sequece v 4l 1, l = 1,..., c 4, c 3. The desired result for γ r,k(p ) ad γ s,k (P ) therefore follows by a combiatio of (8) ad (9). (b) Cosider the case k. It is show first, by cotradictio, that every subpath P (i) k+3 : v i (mod ) v i+k+ (mod ) of legth k+3 withi P : v 0 v 1 v 1 cotais at least k + 1 vertices from V (0) 1 for ay k FSDF f (0) = (V (0) 1 ). Suppose, to the cotrary, that there exists such a subpath P (i) k+3 of P cotaiig oly k vertices from V (0) 1 (ad hece 3 vertices from V (0) 0 ). There is exactly oe possible case: Ú Ü Ú Ý Ú Þ ßÞ Ð ßÞ Ð ßÞ Ð ßÞ Ð Here dark vertices deote elemets of V (0) 1 ad a, b, c, d 0, with a + b + c + d = k. The sequeces of moves ad f (j+1) = move(f (j), v z 1 j v z j ), j = 0,..., c 1 f (c+l+1) = move(f (c+l), v x+l+1 v x+l ), l = 0,..., b 1 reder usafe cofiguratios i P after b + c moves, because v y V (b) 0 is ot adjacet to ay u V (b) 1. This cotradictio shows that V (P (i) k+3 ) V (0) 1 k + 1 for all i = 0, 1,..., 1. I order to fulfil this property, it follows that V (0) 1 (k + 1) k+1 k+3 + r, where r k+3c, rederig the lower boud k + 1 γs,k(p ) (k + 1) + k + 3 k + 3 c, if k, (10)

12 1 with c (mod k + 3). To see that this boud is sharp, partitio the path P ito (j) k+3 subpaths P k+3 : v j(k+3), v j(k+3)+1,..., v j(k+3)+k+ (j = 0,..., k+3 1) ad oe subpath P c : v /(k+3) (k+3),..., v 1 of legth c (mod k + 3). Cosider the safe guard fuctio f (0) = (V (1) 1 ), where P (j) (0) k+3 V 1 = {v i : i 1,,..., k, k+1 (mod k+3)}, j = 0,..., 1, k + 3 where P c V (0) 1 = { {vi : i 0, 1,..., c 1 (mod k + 3)} if 1 c k+ {v i : i 1,,..., c 1 (mod k + 3)} if k+ < c k + ad where V (0) 0 = V (P )\V (0) 1. Clearly f (0) is a k FSDF for P, ad hece γs,k(p ) w(f (0) ) (11) c + 1 = (k + 1) + (c + 1) k + 3 k + 3 (c + 1)(k + 3) = (k + 1) + c + 1 k + 3 k + 3 k + 3 k + k + 3 k + 1 = (k + 1) + k + 3 k + 3 c if k. (1) It follows by (10) ad (1) that k + 1 k + 1 γs,k(p ) = (k + 1) + k + 3 k + 3 c = k + 3, if k, where the last equality ca be proved by rewritig c i terms of ad k. Fially ote that k + 1 k + 3 = 1 if k =. It follows by Propositio 3 that for ay k. But certaily γ s,k(p ) γ s, (P ) = 1 (13) γ s,k(p ) 1 (14) for all k N. A combiatio of (13) ad (14) yields the desired result γs,k (P ) = 1 for all k.

13 13 Note that the correspodig case k = 1 i Theorem 3(b) was established i Theorem 1. Fially, we have the followig cojecture. Cojecture 1 γ r,k (P ) = γ s,k (P ) for ay N. 4. Cycles For cycles it is also possible to establish three of the fiite order parameter values exactly. Theorem 4 For ay cycle C, k + 1 (a) γ r,k (C ) = γ s,k (C ) = 4k + 3, for all k N 0, { (b) γs,k(c k+1 ) = k+3, if 0 k 3, if k 3. Proof: (a) The proof of this result is idetical to that of Theorem 3(a). (b) It ca be show, by exactly the same cotradictio argumet as i Theorem 3(b), that k + 1 γr,k(c ) (k + 1) + k + 3 k + 3 c, if k 3, (15) with c (mod k + 3). To see that this boud is sharp, partitio the cycle C ito (j) k+3 subpaths P k+3 : v j(k+3), v j(k+3)+1,..., v j(k+3)+k+ (j = 0,..., k+3 1) ad oe subpath P c : v /(k+3) (k+3),..., v 1 of legth c (mod k + 3). Cosider the safe guard fuctio f (0) = (V (1) 1 ), where V (C ) V (0) 0 = {v i : i k/ +, k+1 (mod k+3)}, j = 0,..., 1, k + 3 ad where V (0) 1 = V (C )\V (0) 0. Clearly f (0) is a k FSDF for C, ad hece γs,k(c ) w(f (0) ) (16) (k + 1) k+3 + c, if 0 c k + 1 (k + 1) k+3 + c 1, if k + < c k + k + 1 = k + 3 (17)

14 14 for all k 3, exactly as i the proof of Theorem 3(b). The desired result for k 3 therefore follows by a combiatio of (15) ad (17). Fially ote that k + 1 k + 3 = if k = 3. It follows by Propositio 3 that for ay k 3. But certaily γ s,k(c ) γ s, 3(C ) = (18) γ s,k(c ) (19) for all k N. A combiatio of (18) ad (19) yields the desired result γs,k (C ) = for all k Complete bipartite graphs I this sectio we cosider complete bipartite graphs ad fid values for γ s,k ad γs,k for this simple graph class. Theorem 5 For the complete bipartite graph K p,q, 4, k = 1 ad p 4 γ s,k (K p,q ) = γs,k(k (k + 1), 1 < k p p,q ) = p, p + 1 k < p q, k p where p, q N, with p q. Proof: It was show i [3] that γs,1(k p,q ) = 4 if p 4. I order to establish the other three cases, let P ad Q deote the partite sets of K p,q, with P = p ad Q = q. First, cosider the case 1 < k p. Note that f (0) = (V (0) 1 ) is a k FSDF if V (0) 1 P = k +1, V (0) 1 Q = k +1 ad V (0) 0 = V (K p,q )\V (0) 1 (as show i Figure 4.1(a)). Hece p γ s,k (K p,q ) γs,k(k p,q ) (k + 1) if 1 < k (0) by Theorem. Now assume that γ s,k (K p,q ) < (k + 1), the either V (0) 1 P k or V (0) 1 Q k. Assume, without loss of geerality, that V (0) 1 Q k. The o move sequece of the form f (i+1) = move(f (i), u i v i )

15 15 reders a safe guard fuctio f (k) for ay sequece of vertices u i P if v i Q (i = 0,..., k 1), sice { } p p 1 if p is eve + 1 = < p. p if p is odd This cotradictio shows that p γs,k(k p,q ) γ s,k (K p,q ) (k + 1) if 1 < k. (1) The secod case of the theorem therefore follows by a combiatio of (0) ad (1). Now, cosider the case p + 1 k < p ad let f (0) = (V (0) 1 ) be a k FSDF with V (0) 1 = l < p 1. Suppose Q V (0) 1 = c ad P V (0) 1 = l c for some 0 c l. Deote the elemets of the o empty set P V (0) 0 by {v 0,..., v p l+c 1 }, ad cosider the problem vertex sequece v i (i = 0,..., c 1). Clearly ay move sequece of the form move(f (i), u i v i ) with u i Q V (i) 1 (i = 0,..., c 1) will reder a usafe guard fuctio f (c) i K p,q. This cotradictio shows that γ s,k(k p,q ) γ s,k (K p,q ) p if p + 1 k < p. () To see that this boud is sharp, cosider the k FSDF f (0) = (V (0) 1 ) depicted i Figure 4.1(b), where dark vertices deote elemets of V (0) 1, by which it follows that γ s,k (K p,q ) γ s,k(k p,q ) p if p + 1 k < p. (3) The third case of the theorem therefore follows by a combiatio of () ad (3). Fially, cosider the case where k p ad suppose f (0) = (V (0) 1 ) is a p FSDF with V (0) 1 = l < q. Suppose P V (0) 1 = d ad Q V (0) 1 = l d for some 0 d l. Deote the elemets of the o empty set Q V (0) 0 by {v 0,..., v q l+d 1 }, ad cosider the problem vertex sequece v i (i = 0,..., d 1). Clearly ay move sequece of the form move(f (i), u i v i ) with u i Q V (0) 1 (i = 0,..., d 1) will reder a usafe guard fuctio f (d) i K p,q. This cotradictio shows that γ s,p (K p,q ) q. Hece But certaily γ s,k (K p,q ) q if k p. (4) γ s,k (K p,q ) q. (5) Hece the fourth case of the theorem follows by a combiatio of (4) ad (5).

16 16 ½ ÚÖØ È (a) 1 < k É ½ ÚÖØ p (b) Ô ÚÖØ È É p + 1 k < p È É (c) k p Õ ÚÖØ Figure 4.1: Foolproof k secure domiatig fuctios f (0) = (V (0) 1 ) for the complete bipartite graph K p,q (p q). Dark vertices deote elemets of V (0) 1. 5 Coclusio I this paper the previously studied otios of secure domiatio ad of weak Roma domiatio were geeralised i the sese that safe cofiguratios i a simple graph were ot merely sought after oe move, but rather after k 1 moves. Some geeral properties of these geeralised domiatio umbers were established i 3, after which the parameter values were foud for certai simple graph structures i 5. There is ample scope for the determiatio of parameter values for specific graph structures, such as the values of γ r,k (P ) ad γ r,k (C ), the various parameter values for complete multipartite graphs, etc. Further work may ivolve a umber of iterestig geeralisatios: (i) I our work the problem vertex sequece was always kow completely i advace by the strategist. However, the situatio where these problem vertices are made kow (ad are dealt with) oe at a time might be a more realistic sceario i terms of games of strategy, ad this geeralisatio deserves to be ivestigated. (ii) It might also be worth while allowig for a umber of cosecutive moves before requirig the graph to be protected, istead of requirig safe cofiguratios after each move (similar to the watchma s walk problem described i [4]). Ackowledgemets A.P. Burger, E.C. Cockaye & C.M. Myhardt were visitig the Departmet of Applied Mathematics at Stellebosch Uiversity whe research towards this paper was doe. The paper is based upo work supported by

17 17 the South Africa Natioal Research Foudatio uder grat umber GUN ad Research Sub Committee B at the Uiversity of Stellebosch. Ay opiios, fidigs ad coclusios or recommedatios expressed i this paper are those of the authors ad do ot ecessarily reflect the views of the Natioal Research Foudatio. Refereces [1] E.J. Cockaye, P.A. Dreyer, S.M. Hedetiemi ad S.T. Hedetiemi, Roma domiatio i graphs, to appear i Discrete Math. [] E.J. Cockaye, O. Favaro ad C.M. Myhardt, Secure domiatio, weak Roma domiatio ad forbidde subgraphs, to appear i Bull. Ist. Combi. Appl. [3] E.J. Cockaye, P.J.P. Grobler, W.R. Grüdligh, J. Mugaga ad J.H. va Vuure, Protectio of a graph, to appear i Utilitas Math. [4] B.L. Hartell, D.F. Rall ad C.A. Whitehead, The watchma s walk problem: A itroductio, Cog. Numer., 130 (1998), [5] T.W. Hayes, S.T. Hedetiemi ad P.J. Slater, Fudametals of Domiatio i Graphs, Marcel Dekker, New York, [6] M.A. Heig ad S.M. Hedetiemi, Defedig the Roma Empire A ew strategy, to appear i Discrete Math. [7] I. Stewart, Defed the Roma Empire!, Scietific America, December 1999,