Tight bounds for eternal dominating sets in graphs

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1 Discree Mahemaics Tigh bounds for eernal dominaing ses in graphs John L. Goldwasser a, William F. Klosermeyer b a Deparmen of Mahemaics, Wes Virginia Universiy, Morganown, WV 6506, USA b School of Compuing, Universiy of Norh Florida, Jacksonville, FL , USA Received 9 January 007; received in revised form 1 June 007; acceped 6 June 007 Available online 16 June 007 Absrac The eernal dominaion number of a graph is he number of guards needed a verices of he graph o defend he graph agains any sequence of aacks a verices. We consider he model in which a mos one guard can move per aack and a guard can move across a mos one edge o defend an aack. We prove ha here are graphs G for which γ G αg+1, where γ G is he eernal dominaion number of G and αg is he independence number of G. This maches he upper bound proved by Klosermeyer and MacGillivray. 007 Elsevier B.V. All righs reserved. Keywords: Dominaing se; Eernal dominaing se; Eernally secure se; Kneser graph 1. Inroducion We consider he problem of defending he verices in a finie, undireced graph from a sequence of aacks [,5,7]. In his problem, guards are locaed a verices, can proec he verices a which hey are locaed, and can move o a neighboring verex o defend an aack here. This paper deals wih he eernal version of he problem in which he sequence of aacks can be infiniely long. Furher, we will allow a mos one guard o move o defend each aack. Oher variaions of his kind of problem have also been sudied, including Roman Dominaion [3,6], Weak Roman Dominaion [4], k-secure ses [1], and eernal m-secure ses [5]. The erm Roman Dominaion sems from he problem s ancien origins in Emperor Consanine s effors o defend he Roman Empire from aackers [6,10 1]. These problems relevance o miliary sraegy is discussed in [9,11]. The open neighborhood of verex v is denoed by Nv, and is closed neighborhood Nv {v} is denoed by N[v]. A dominaing se of G is a se D V such ha, for all v, N[v] D =. Le R = r 1,r,... be an arbirarily long sequence of verices of a graph. The elemens of R are he locaions of a sequence of consecuive aacks a verices, each of which mus be defended by a guard. A se D is an eernal dominaing se also known as an eernally secure se, if for all possible sequences of aacks R = r 1,r,...here exiss a sequence D = D 1,D...of dominaing ses and a sequence of verices v 1,v,...such ha D i+1 = D i \{v i } {r i }, where v i D i N[r i ]. Noe ha v i = r i is possible. The se D i+1 is he se of locaions of he guards afer he aack a r i is defended. If v i = r i, we say ha he guard a v i has moved o r i. address: klosermeyer@homail.com W.F. Klosermeyer X/$ - see fron maer 007 Elsevier B.V. All righs reserved. doi: /j.disc

2 590 J.L. Goldwasser, W.F. Klosermeyer / Discree Mahemaics Someimes we model he evolving sequence D 1,D,...as a wo-player game: Player 1 chooses D 1 and he verices v 1,v,...while Player chooses he verices r 1,r,...Player 1 chooses v i o defend he aack Player makes a r i. The size of a smalles eernal dominaing se in G is he eernal dominaion number also known as he eernal securiy number and is denoed by γ G or simply γ []. In his paper, we prove here exis graphs G for which γ G αg+1, where αg is he size of he larges independen se in G. This improves resuls from [7]. The original problem of bounding he eernal dominaion number by a funcion of he independence number is from Goddard e al. [5], who asked wheher he eernal dominaion number of a graph can be bounded by a consan imes is independence number. Since i is known from Klosermeyer and MacGillivray [7] ha γ G αg+1, for all graphs G, his bound is igh. Cerain large complemens of Kneser graphs are used o obain our resul. In addiion, we deermine he eernal dominaion number of he line graph of K n which is isomorphic o he complemen of a Kneser graph, using inducion in an unusual way.. Background Le θg denoe he clique-covering number of G, i.e., he chromaic number of he complemen of G. Goddard e al. [5] poined ou ha, for all graphs G, αg γ G θg. To see he lefmos inequaliy, imagine a sequence of consecuive aacks a independen verices. To see he righmos inequaliy, observe ha a single guard can eernally defend all verices of a clique. Goddard e al. [5] also proved ha if α = hen γ 3, and conjecured ha here is a consan c such ha γ G c for all graphs wih αg = 3. For each posiive ineger, le F be he larges posiive ineger r such ha here exiss a graph G wih αg = and γ G = r. In[7], i was shown ha 3 F. 1 To see he lower bound, noe ha αc 5 = and γ C 5 = 3, where C 5 is he 5-cycle. Then n disjoin C 5 s form a graph wih independence number n and eernal dominaion number 3n. If you wan o consruc a conneced graph wih he same propery, simply add a verex adjacen o all he ohers. In his paper, we show ha F= for each. The fac ha equaliy acually holds may be a bi surprising in view of he proof of he upper bound in 1 given in [7] which shows roughly ha any graph G wih αg = can be eernally dominaed wih guards so ha a all imes he guards can be pariioned ino independen ses of sizes 1,,..., he proof allows some of hese ses o be empy. Wihou requiring his exra srucure, we show ha cerain graphs sill canno be eernally dominaed by fewer han guards. 3. Tigh bound on eernal dominaion number Le Gn, k be he graph wih verex se equal o he se of all k-subses of an n-se and where wo verices are adjacen if and only if heir inersecion is non-empy so i is he complemen of a Kneser graph. We refer o he elemens of he n-se as symbols and use he inegers 1,,...,no represen hese symbols. To illusrae Gk + k 1,k, when = 1 and k =, he graph is G3, which is simply K 3. Similarly when = and k = we ge G5,, which has 10 verices ha we could label 1,, 1, 3,...,4, 5 and edges such as 1,, 1, 3. However, for our purposes, k will need o be fairly large relaive o, as we describe below. Theorem 1. For each posiive ineger, if k is sufficienly large, hen he graph Gk +k 1,khas eernal dominaion number. From inequaliy 1 we hen ge he following.

3 J.L. Goldwasser, W.F. Klosermeyer / Discree Mahemaics Theorem. For each posiive ineger, if αg =, hen γ G holds. and here exiss a graph for which equaliy Proof of Theorem 1. We will show ha if Gk + k 1,kcan be eernally dominaed wih fewer han guards, hen here is an upper bound on k. We use L i o denoe he se of symbols associaed wih a verex of V Gk+k 1,k. We abuse noaion and also use L i o refer o he verex iself, when clear from he conex. Suppose Player 1 iniially places he guards a verices L 1,L,...,L which consiues a dominaing se of he graph. We le M be he union as a mulise of L 1,L,...,L 1. Le p i be he number of symbols in {1,,...,k+ k 1} ha have mulipliciy i in M i = 0, 1,,... Since {L i } is a dominaing se, p 0 <k. Therefore, k i 1 p i k + k 1 1 and ip i = k i Suppose ha wih he specified iniial configuraion of guards he graph can be eernally dominaed wih 1 guards. We also assume ha of all iniial configuraions of he 1 guards so ha he graph can be eernally dominaed, we have one in which he symbol mulipliciy vecor p 0,p 1,p,...is lexicographically las so p 0 is as large as possible, and of all hese, p 1 is as large as possible, and of all hese p is as large as possible, and so on. For each posiive ineger i, le Q i be he se of all L j s ha have a symbol of mulipliciy i bu no symbol of smaller mulipliciy and le q i = Q i. For each i and each L j in Q i, arbirarily choose a symbol of mulipliciy i and call i special he same symbol could be special in more han one L i. We claim ha p i q i k 1, i = 1,, 3, Suppose no. Le r be he smalles ineger i for which inequaliy 4 does no hold. This means here are a leas k symbols of mulipliciy r ha are no special. Player aacks a verex L u conaining any k of hem, Player 1 can only defend he aack wih a guard from a verex L j in Q s for some s r. Wih L j replaced by L u, in he union as a mulise of he new se of guarded verices, each symbol in L j \L u has is mulipliciy decreased by 1 while each symbol in L u \L j has is mulipliciy increased by 1. This means he symbol ha was special in L j now has mulipliciy s 1 and perhaps oher symbols also had heir mulipliciies reduced from s o s 1, bu he only symbols ha have increased mulipliciy in he new se of guarded verices now have mulipliciy s + 1. This implies ha he mulipliciy vecor p 0,p 1,p,...for he new configuraion of guards has p i =p i for i s and p s 1 >p s 1, conradicing he assumpion ha p 0,p 1,p,...is las in he lexicographic ordering of mulipliciy vecors of mulise unions of ses of 1 guards in a configuraion in which hey would be able o eernally dominae he graph. Hence, inequaliy 4 holds as claimed. From and 3 we have k k 1 p i ip i i 1 i 1 = ip i + ip i i iq i + k 1 = k 1 + iq i.

4 59 J.L. Goldwasser, W.F. Klosermeyer / Discree Mahemaics Hence, [ k = k Therefore, if k> 4 /4, ] + 1 iq i i [ ] = < guards are no enough o eernally defend he graph. The bound 4 /4 could be improved because o derive i we used q i 1, whereas in fac he sum of all he q i s has his bound. We remark ha if k> 4 /4, he graph Gk + k 1,kwill be large even for small. For example, if = 4 and k = 65, he graph has more han verices. Of course, he eernal dominaion number is also equal o for some values of k less han 4 /4. We have shown ha if = 3, hen γ Gk + k 1,k= for all k 4 he proof is omied, bu uses similar echniques as above. 4. Bounds for line graphs of K n As saed above, he graph Gn, k is he complemen of he Kneser graph Kn, k. Hence θgn, k = n k +, because of he celebraed Lovász heorem ha saes ha his is he chromaic number of Kn, k a direc proof when k = is easy. So γ Gn, θgn, = n. We show ha equaliy holds and noe ha Gn, is isomorphic o he line graph of K n. Theorem 3. γ Gn, = αgn, 1 ifnisodd. γ Gn, = αgn, ifniseven. Proof. We hink of each verex x, y of Gn, as he edge [x,y] of K n. So a dominaing se in Gn, corresponds o a se of edges in K n ha span n 1orn verices. From his viewpoin, Player 1 chooses such a se D of edges, Player aacks an edge e of K n no in D, Player 1 has o delee an edge of D inciden o D, and so on, wih Player winning if afer Player 1 has deleed an edge, less han n 1 verices are spanned by he remaining edges because hen Player can aack an edge no inciden o any in he se. We now show n 3 guards are no enough o defend Gn,. Suppose Player 1 chooses n 3 edges. A graph wih n 3 edges on n verices has a leas hree componens ha are rees possibly wih one verex. We show ha by a sequence of aacks, Player can force a ransformaion of wo of hese componens ino isolaed verices. Choose any wo of hese hree ree componens, say T 1 and T. Call a verex a pendan verex if i has degree 1. We show by inducion on he minimum number of non-pendan verices in T 1 and T ha by a sequence of aacks, Player can force a ransformaion o a pair of rees, one of which is an isolaed verex. If he minimum number of non-pendan verices in T 1 and T is 0 or 1, hen one of hem, say T 1 is a sar. Le x be is apex and le y be any pendan verex of T. Player aacks [x,y]. Player 1 eiher defends wih an edge of T 1, leaving an isolaed verex, or wih he edge of T conaining y, which leaves a sar T 1 wih one more edge han T 1 and a ree T wih one less edge han T. Repeaing his procedure evenually reduces T o a single edge. One more aack leaves an isolaed verex. Now assume T 1 has m non-pendan verices and T has a leas his many for some m. Player aacks [x,y] where x is any non-pendan verex of T 1 and y is any pendan verex of T. Choice 1 for Player 1 is o defend wih an edge of T 1, say [x,w]. Then we are lef wih rees T 1 and T where x is a verex of T, w is a verex of T 1 and T 1 has fewer non-pendan verices han T 1. Choice for Player 1 is o defend wih he edge of T conaining y, leaving rees T 1 and T where T 1 has he same number of non-pendan verices as T 1 and T has one fewer edge han T. Repeaing

5 J.L. Goldwasser, W.F. Klosermeyer / Discree Mahemaics his procedure evenually forces Player 1 o use choice 1 above, leaving a ree T 1 wih fewer non-pendan verices han T 1, and evenually reducing T 1 o a sar, and hen o an isolaed verex u. Now repea his process wih he oher wo ree componens no u. This evenually resuls in wo rees, one of which is an isolaed verex, say v. Then [u, v] is no defended. 5. Fuure direcions I was shown in [8] ha γ = θg for cacus graphs, powers of cycles, and complemens of powers of cycles. I is likely ha he same is rue for oher ineresing classes of graphs. Problem 1. Deermine igh bounds on γ G when G belongs o some paricular class of graphs e.g., ouerplanar graphs, planar graphs. Noe ha Problem 1 is meaningful only for classes conaining graphs ha are no perfec, because he Perfec Graph Theorem ensures ha γ G = θg = αg for all perfec graphs. However, here exis graphs ha are no perfec whose eernal dominaion number equals heir independence number. Baranyai s heorem, see for example [13], says ha if k divides n, hen he complee k-graph on n verices has a 1- facorizaion. Hence, θkn, k = n k n/k = n 1 = αkn, k. k 1 This means if k divides n hen γ Kn, k = αkn, k while here is a huge gap beween γ Gn, k and αgn, k. To some exen, his huge gap seems due o he srucure of he induced odd cycles in Gn, k, hough is complemen Kn, k also has many odd cycles and no gap a all. We noe ha all known graphs G wih θg = αg + 1haveγ G = θg. We do know which graphs have dominaion number equal o eernal dominaion number. Le γg denoe he size of a smalles dominaing se of G. Theorem 4 Klosermeyer and MacGillivray [8]. γg = γ G if and only if γg = θg. However, i remains o characerize which graphs have eernal dominaion number equal o independence number. Acknowledgmens We hank he anonymous referees for heir helpful commens. References [1] A. Burger, E. Cockayne, W. Gründlingh, C. Mynhard, J. van Vuuren, W. Winerbach, Finie order dominaion in graphs, J. Combin. Mah. Combin. Compu [] A. Burger, E. Cockayne, W. Gründlingh, C. Mynhard, J. van Vuuren, W. Winerbach, Infinie order dominaion in graphs, J. Combin. Mah. Combin. Compu [3] E. Cockayne, P. Dreyer, S.M. Hedeniemi, S.T. Hedeniemi, Roman dominaion in graphs, Discree Mah [4] E. Cockayne, O. Favaron, C. Mynhard, Secure dominaion, weak roman dominaion and forbidden subgraphs, Bull. Ins. Combin. Appl [5] W. Goddard, S.M. Hedeniemi, S.T. Hedeniemi, Eernal securiy in graphs, J. Combin. Mah. Combin. Compu [6] M. Henning, S.M. Hedeniemi, Defending he Roman empire a new sraegy, Discree Mah [7] W. Klosermeyer, G. MacGillivray, Eernal securiy in graphs of fixed independence number, J. Combin. Mah. Combin. Compu., 007, o appear. [8] W. Klosermeyer, G. MacGillivray, Eernal dominaing ses in graphs, J. Combin. Mah. Combin. Compu., 007, o appear. [9] D. Ochmanek, Time o resrucure US defense forces, Sci. Technol. Winer [10] I. Peersen, Defending he Roman empire, MahTrek, Sepember 11, [11] C.S. ReVelle, K.E. Rosing, Defendens imperium Romanum: a classical problem in miliary sraegy, Amer. Mah. Monhly [1] I. Sewar, Defend he Roman empire! Scienific American, December 1999, pp [13] J.H. van Lin, R.M. Wilson, A Course in Combinaorics, Cambridge Universiy Press, Cambridge, 006.