FIGHTING CONSTRAINED FIRES IN GRAPHS

Transcription

1 FIGHTING CONSTRAINED FIRES IN GRAPHS ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT Abstract The Firefighter Problem is a simplified model for the spread of a fire or disease or computer virus) i a etwork A fire breaks out at a vertex i a coected graph, ad spreads to each of its uprotected eighbours over discrete time-steps A firefighter protects oe vertex i each roud which is ot yet bured While maximizig the umber of saved vertices usually requires a strategy o the part of the firefighter, the fire itself spreads without ay strategy We cosider a variat of the problem where the fire is costraied by spreadig to a fixed umber of vertices i each roud I the two-player game of k-firefighter, for a fixed positive iteger k, the fire chooses to bur at most k uprotected eighbours i a give roud The k-survivig rate of a graph G is defied as the expected percetage of vertices that ca be saved i k-firefighter whe a fire breaks out at a radom vertex of G We supply bouds o the k-survivig rate, ad determie its value for families of graphs icludig wheels ad prisms We show usig spectral techiques that radom d regular graphs have k-survivig rate at most +Od / )) k+ We cosider the limitig survivig rate for coutably ifiite graphs I particular, we show that the limitig survivig rate of the ifiite radom graph ca be ay real umber i [/k + ), ] Itroductio The Firefighter Problem was itroduced by Hartell [5], ad is a simplified determiistic model of the spread of fire, diseases, ad computer viruses i graphs All graphs we cosider are coected, fiite except i Sectio 4), simple, ad udirected I Firefighter, vertices are either burig or ot There is oe firefighter who is attemptig to cotrol the fire Oce a vertex is occupied by the firefighter, it ca ever bur i ay subsequet roud, ad is called protected The fire begis at some vertex i the first roud, ad the firefighter chooses some vertex to save The firefighter ca visit ay vertex i a give roud for example, he ca jump betwee two o-joied vertices from oe roud to the ext), but caot protect a vertex o fire The fire acts without itelligece, ad spreads to all o-protected eighbours; such vertices are called bured Oce a vertex has bee protected, its state caot chage; that is, it ca ever be o fire The process stops whe the fire ca o loger spread A vertex is saved if it is ot bured at the ed of the process Firefighter has bee a cosidered i several familiar graph classes, such as fiite ad ifiite grids [7, 9], cubic graphs [8], ad trees [8, 6] The problem has bee studied from both structural ad algorithm viewpoits; see the survey [] for additioal backgroud We cosider a variat of firefightig, called k-firefighter for a fixed positive iteger k, where i each roud, the fire chooses at most k eighbourig vertices to bur I k-firefighter, a move for the fire is to spread to at most k uprotected vertices The reader should ote that this results i a two-player game, where a optimal strategy for the fire aims to bur as 99 Mathematics Subject Classificatio 05C57, 05C80, 05C63 Key words ad phrases Firefighter, survivig rate, radom regular graphs, ifiite radom graph Supported by grats from NSERC, MITACS, ad Ryerso

2 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT may vertices as possible, ad a optimal strategy for the firefighter aims to save as may vertices as possible By keepig a score betwee the vertices bured ad the vertices saved o a fiite graph), k-firefighter ca also be iterpreted as a combiatorial game We ote that the game of k-firefighter was first suggested as a directio for future ivestigatio i [9] The game of k-firefighter cosiders the spread of fire with limited resources Rather tha spreadig to every eighbour, the fire is costraied to spread to a fixed umber of them Although this may be appear less atural tha the applicatios of the origial Firefighter problem, it has potetial applicatios to the spread of gossip, ews, or iformatio i a social etwork: agets i such etworks have fiite time ad resources, ad spread the gossip to a fiite selectio of frieds or followers For the rest of the paper, k is a fixed positive iteger We ote that ulike Firefighter, where the fire has o strategy, the choice of strategy for the fire is importat i k-firefighter Cosider the graph G i Figure, where m 3 Suppose the fire breaks out at x ad the firefighter protects a I Firefighter, i the secod roud the fire spreads to both y ad b The firefighter ca the oly save two vertices of G However, i -Firefighter if the fire chooses to burs y i the secod roud, the the firefighter ca save all of K m by protectig b If the fire burs b rather tha y i the secod roud, the the firefighter ca save m vertices i K m We ote, however, that i some a x y b K m Figure The graph G graphs for example, i a clique or a path), the outcome of the game does ot deped o how the fire spreads The survivig rate ρg) of a graph G was itroduced by Cai ad Wag [7] ad defied as the expected proportio of vertices that ca be saved whe a fire breaks out at a vertex of G chose uiformly at radom Exact values for the survivig rate have bee determied for paths ad cycles [7] ad bouds have bee determied for trees [7, 8], plaar graphs [7], K 4 - free mior graphs [], ad outerplaar graphs [8] For sparse graphs, survivig rates should be relatively large Fibow, Wag, ad Wag [] showed that ay graph G with vertices ad at most 4 ε) edges has the property that ρg) 6 5 ε, where 0 < ε < is fixed I [9] the third author of this paper improved this to show that for graphs with size at most 5 ε) has survivig rate ρg) ε, where 0 < ε < is fixed Moreover, a 60 costructio of a radom graph has bee proposed to show that o further improvemet is possible; that is, 5 is the threshold see also [0] for a atural extesio of this threshold) We ow defie the survivig rate of graphs i the game of k-firefighter For a vertex v i G, defie s k G, v) to be the umber of vertices that ca be saved if a fire breaks out at v

3 FIGHTING CONSTRAINED FIRES IN GRAPHS 3 For a fiite graph G, defie its k-survivig rate to be ρg, k) = s k G, u) u V G) Note that s k G, u)/ is the proportio of vertices that ca be saved whe a fire breaks out at u Thus, ρg, k) is the expected percetage of vertices that ca be saved whe a fire breaks out at a radom vertex of G uiform distributio is used for the iitial placemet) For example, for a clique For a path, ρk, k) = )/k + ) ) k + ρp, k) = ρp ) = + = + To derive these bouds, ote that i a clique, the firefighter ca save oe vertex i each step of the game ad the the fire spreads to some k vertices, o matter where the fire breaks out After )/k + ) steps the process is fiished For a path, the firefighter ca save vertices if the fire breaks out at the ed-vertices, ad vertices otherwise) We focus o the computig the survivig rate for k-firefighter i various cotexts I Sectio, we supply a upper boud i Theorem for ρg, k), where G is a coected graph I Theorems ad 3, we give the exact values of the k-survivig rate for wheels ad certai prisms, respectively We show that radom regular graphs have low k-survivig rates for all values of k It is show i Theorem 4 that asymptotically almost surely radom regular graphs have k-survivig rate at most + d / k + O)) k + = + Od / )) k + where d is the degree of regularity This teds to as d teds to ifiity, which is the k+ smallest possible survivig rate i k-firefighter see )) We fiish with Sectio 4, where we itroduce the k-survivig rate for ifiite graphs as the limit of the k-survivig rate of a chai of fiite coected graphs The defiitio i the ifiite case allows i certai cases) for differet survivig rates For some graphs, such as cliques or paths, all chais of coected graphs give the same limitig k-survivig rate We prove i Theorem 9 the surprisig result that the k-survivig rate of the ifiite radom graph ca, i fact, be ay real umber i [/k + ), ] We emphasize that while the choice by the fire to spread to k eighbours may greatly ifluece the outcome of the game as the example i Figure illustrates), i may cases, it does ot Ideed, we use the itelligece of the fire i our proofs below explicitly oly i Lemma 0 Bouds ad graph families As addig edges does ot icrease the k-survivig rate, it follows that cliques have the smallest survivig rates Hece, for a graph with vertices we have that )/k + ) ) ρg, k) ) k +

4 4 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT We ow derive a upper boud for the k-survivig rate of a coected graph as a fuctio of k ad its order Theorem For a coected graph G o vertices, ρg, k) + + ) k + ) ) + O k + Note that the boud i ) is sharp as equality holds for a star o vertices Proof Sice ρg, k) ρg e, k) for e EG), we may assume without loss of geerality that G is a tree For a positive iteger i, defie [i] = {,,, i} Let d, d,, d ) be the degree sequece for T, where d i d i+ for i [ ] Let v i be the vertex with degree d i for i [] If a fire starts at v i, the the firefighter ca save at most oe of every k + eighbours of v i, thereby savig at most d i eighbours of v k+ i Thus, the fire burs at ) least + d i d i vertices, which gives the followig boud o the proportio of vertices k+ bured: ρt, k) As i= d i = ), we have that i= + d i ρt, k) 3 di k + ) di ) k + i= = di k + We leave it as a exercise to show that a degree sequece of trees which maximizes i= di k + is,,,,, ) Therefore, we have that i= ρt, k) = k + = + + k + ) k + To further illustrate the parameter ρg, k), we ow cosider exact k-survivig rates for some families of graphs Let W deote the wheel graph, cosistig of a )-cycle with all vertices of the cycle joied to a cetre vertex

5 Theorem FIGHTING CONSTRAINED FIRES IN GRAPHS 5 ρw, k) = / if k = ad = if k = ad 5 /4 if k ad = 4 /5 if k = ad = 5 9/5 if k > ad = k+ if k ad 6 Proof The cases for = 4, 5 are straightforward ad so omitted Assume that 6 If the fire breaks out at oe of the vertices of degree three, the the firefighter should protect the cetre vertex What remais for the fire is the cycle C If k =, the the firefighter ca save all but three vertices o C If k =, the the firefighter ca save all but four vertices o C If the fire breaks out at the cetre vertex, the firefighter ca save exactly oe of every k + eighbours of the cetre vertex, savig a total of vertices k+ Thus, we have that ρw, ) = 3) ) + k + ρw, k) = 4) ) + k + for k ) = ) = k+ The Cartesia product of G ad H, writte G H, has vertex set V G) V H) Vertices a, b) ad c, d) are joied if a = c ad bd EH), or ab EG) ad b = d We cosider prisms which are Cartesia products where oe of the factors is K Theorem 3 For k, we have the followig ) ρp K, ) = + 3 for 4; ) ρp K, k) = for 6; 3) ρc K, ) = for 4; 4) ρc K, k) = 3 for 5 Proof The cases for = 4, 5 are trivial ad so omitted; assume without loss of geerality that 6 I additio, the cases whe k = that is, items ) ad 3)) are proved i a maer aalogous to the cases whe k, ad so are omitted First, cosider P K Due to symmetry, there are four possible cases to cosider, depedig o where the fire breaks out iitially These are described by Figure 3 where the vertices bured are deoted by a black square vertex ad the vertices protected are deoted by a white roud vertex the umber ext to a vertex idicates the roud at which the vertex was protected or bured) The iitial bured vertex is deoted by the large square Figure supplies a optimal strategy for the firefighter Coutig the umber of vertices saved i each of the four cases, as well as the umber of times each case ca occur, we have ρp K, k) = ) 4 ) + 4 4) + 4 5) + ) 6) 4 = 3 + 7

6 6 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT Case Case Case 3 Case 4 Figure The four cases for P K, where k Cosider C K for 5 ad k Due to symmetry, there is oly oe possible situatio to cosider: Case 4 The 6 vertices are saved, supplyig that ρc K, k) = ) ) 6) = Radom d regular graphs are flammable We ow cosider k-firefighter played o radom d-regular graphs with uiform probability distributio The probability space is deoted G,d A evet holds asymptotically almost surely or aas i G,d if it holds with probability tedig to oe for with d fixed, with the proviso that eve if d is odd As we will see i Theorem 4, radom regular graphs are flammable, i the sese that the fire ca aas bur a sizeable proportio of the graph Istead of workig directly i G,d, we use the pairig model of radom regular graphs, first itroduced by Bollobás [4] Suppose that d is eve, as i the case of radom regular graphs, ad cosider d poits partitioed ito labeled buckets v, v,, v of d poits each A pairig of these poits is a perfect matchig ito d/ pairs Give a pairig P, we may costruct a multigraph GP ), with loops allowed, as follows: the vertices are the buckets v, v,, v, ad a pair {x, y} i P correspods to a edge v i v j i GP ) if x ad y are cotaied i the buckets v i ad v j, respectively It is a easy fact that the probability of a radom pairig correspodig to a give simple graph G is idepedet of the graph, hece the restrictio of the probability space of radom pairigs to simple graphs is precisely G,d Moreover, it is well kow that a radom pairig geerates a simple graph with probability asymptotic to e d )/4 depedig o d but ot o Therefore, ay evet holdig aas over the probability space of radom pairigs also holds aas over the correspodig space G,d For this reaso, asymptotic results over radom pairigs suffice for our purposes Oe of the advatages of usig this model is that the pairs may be chose sequetially so that the ext pair is chose uiformly at radom over the remaiig uchose) poits For more iformatio o this radom graph model, see [] The adjacecy matrix A = AG) of a give d-regular graph G with vertices has real eigevalues which we deote by d = λ λ λ Defie λ = λg) = max λ, λ ) Hece, λ is the largest absolute value of a eigevalue other tha d for more details, see [3, 7]) We ow preset a asymptotic upper boud for the k-survivig rate of radom d-regular graphs for all values of d ad k

7 FIGHTING CONSTRAINED FIRES IN GRAPHS 7 Theorem 4 Let d 3, k, ad fix ε > 0 Let λ = d + ε The, for G G,d we obtai that aas + o)) ρg, k) + λ )) k d + = + Od / )) k + d d λ k + By ), we have that ρg, k) as d Hece, for large values of d, aas k+ radom d-regular graphs have, i a certai sese, the smallest possible k-survivig rate A stroger but umerical) result is preseted i the ext subsectio Before we prove Theorem 4 we eed a few lemmas, the first of which is related to expasio properties of radom regular graphs The value of λ for radom d-regular graphs has bee studied extesively A major result due to Friedma [4] is the followig Lemma 5 [4]) For every fixed ε > 0 ad for G G,d, aas λg) d + ε The umber of edges ES, T ) betwee sets S ad T is expected to be close to the expected umber of edges betwee S ad T i a radom graph of edge desity d/, amely d S T / A small λ or large spectral gap) implies that this deviatio is small The followig useful boud is essetially proved i [, 3] Lemma 6 Expader Mixig Lemma) Let G be a d-regular graph with vertices ad set λ = λg) The for all S, T V we have that d S T ES, T ) λ S T Note that S T does ot have to be empty; i geeral, ES, T ) is defied to be the umber of edges betwee S \ T to T plus twice the umber of edges that cotai oly vertices of S T For our purpose here it is better to apply a slightly stroger lower estimate proved i [, 3] for ES, V \ S) : d λ) S V \ S 3) ES, V \ S) for all S V The followig lemma was essetially proved i [9], although we iclude a full proof here for completeess For d 3, a cycle is called short if it has legth at most L = log d log d Lemma 7 If d 3 ad G G,d, the aas the umber of vertices that belog to a short cycle is at most log Proof Let u V G) ad let N i u) deote the set of vertices at distace at most i from u A balaced d-regular tree cotais d vertices o the first level, dd ) vertices o the secod level, ad so o Let f i deote the umber of vertices i a balaced d-regular tree with i levels; that is, i f i = + d j=0 d ) j = + dd )i ) d We will show that i the early stages of this process, the graphs grow from u ted to be trees aas; hece, the umber i of elemets i N i u) is equal to f i aas I other words, if we expose the vertices at distace,,, i from u step-by-step, the we have to avoid at

8 8 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT step j edges that iduce cycles That is, we wish ot to fid edges betwee ay two vertices at distace j from u or edges that joi ay two vertices at distace j to a same vertex at distace j + from u We will refer to edges of this form as special Note that the expected umber of special edges at step i+ is equal to O i /) = Ofi /) = Od ) i /) There are O i ) edges created at this poit; for a give edge the probability of beig special is O i /m) Therefore, the expected umber of special edges foud up to step i = L/ is equal to i O d ) j / ) = O d ) i / ) = O log / ) j=0 Recall that L = log d log d ) Hece, the expected umber of vertices that belog to a cycle of legth at most L is Olog ) ad the assertio follows from Markov s iequality We ow prove Theorem 4 Proof of Theorem 4 Cosider the vertex set U cosistig of vertices from G that do ot belog to a short cycle of legth at most L By Lemma 7, sice U log aas, we have that aas ρg, k) = s k G, v) v V G) = s k G, v) + s k G, v) v / U = o) + U v U s k G, v) v U Therefore, it is eough to show that aas for all u U s k G, u) + o)) + λ k + d k + d d λ )) Let us ote that whe d = 3 ad u / U belogs to the triagle, the firefighter ca save the eighbour of u that does ot belog to the triagle i the first roud, ad the apply a greedy strategy of protectig ay vertex adjacet to the fire This strategy saves at least half of vertices, regardless of the value of k For k = it is, i fact, possible to save all but the three vertices that belog to the triagle Therefore, the desired property might ot hold for u / U, but agai, this is ot causig ay problems sice aas almost all vertices are i U) Let u U ad let B t, F t deote the set of vertices bured or protected i roud t, respectively We will examie three successive phases for the fire spreadig from u I the first phase cosistig of rouds up to some costat t 0, the fire spreads to all o-protected vertices adjacet to the fire but fewer tha k ew vertices catch fire i each roud We will use the fact that G is locally a tree to boud t 0 I the secod phase defied as those rouds up to a certai t = t ), the fire spreads to k eighbours, but there are still o short cycles The subgraph iduced by B t is a tree ad there will be + o))kt bured vertices I the third stage after roud t, there are short cycles that clearly help the firefighter; however, may vertices are already bured, which makes it impossible for the firefighter to save too may vertices

9 FIGHTING CONSTRAINED FIRES IN GRAPHS 9 I the first phase, i order to miimize B t the firefighter should use a greedy strategy; that is, protect a vertex adjacet to the vertex o fire O the other had, if k d, the the fire caot use the whole power i this phase ad ca oly spread to less tha k vertices i each roud However, this ca occur oly for a costat umber of iitial steps Ideed, i the first roud the fire spreads to at least d vertices, the to at least d ) ) ew oes, ad so o I roud t durig this phase, there are at least d ) t d ) t d ) t 3 d ) ) ew vertices o fire Therefore, = d ) t d )t d t 0 ) kd ) log d +, d 3d + = d )t d 3d + ) + d so it is a costat that does ot deped o I the secod phase, after roud t 0, the fire is free to spread to k ew vertices i each roud After t = L = log d log d rouds, there are B t = Okt 0 ) + kt t 0 ) = + o))kt vertices o fire that form a tree We ow cosider the third stage after roud t It is straightforward to see that the fire will be spreadig up to roud ˆT whe EB ˆT, V \ B ˆT ) = EB ˆT, F ˆT ); that is, there is o vertex adjacet to the fire that is ot protected Moreover, if for every t 0 < t T we have that EB t, V \ B t ) EB t, F t ) + dk, the fire spreads with the full speed; that is, B t+ = B t + k durig this stage of the process Sice there are at least dk edges from bured to o-protected vertices, there must be at least k o-protected vertices adjacet to the fire) Hece, B t = + o))kt durig this time period Note that from 3) it follows that for ay roud t EB t, V \ B t ) d λ) B t V \ B t = + o)) d λ)kt kt) Sice F T ca receive at most d F T = dt edges, T ca be take to be arbitrarily large as log as T satisfies d λ)kt kt ) + o)) dt + dk = + o))dt Whe T is maximized, T + fails this test which implies that kt + ) + o)) d ) d λ k ad so 3) T + o)) k ) ) d d λ k

10 0 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT A mildly stroger result ca be obtaied by estimatig the umber of edges betwee B t ad F t usig Lemma 6, ad iequality 3), rough lower boud for T EB t, F t ) d B t F t B t + λ F t F t ) dkt = + o)) + λt k We therefore have milder coditio for T d λ)kt kt ) dkt + o)) ad so k + )T + o)) λ d + λt ) k, + k kt Usig 3), we get that T ca be arbitrarily large, provided that T satisfies the followig iequality k + )T + o)) λ ) ) d k + d d λ k As before, whe T is take such that T + fails this slightly milder) test, we have that the third phase lasts util time T ad T + o)) λ ) )) d k + k + d d λ k Hece, the umber of vertices saved s k G, u) is at most + o))kt + o)) k + ad the assertio follows from Lemma 5 + λ d ) )) k d +, d λ We ote that for the essetial elemets used i the proof of Theorem 4 are Lemmas 5, 7, ad 3) The proof will go through i the exact same fashio for a family of radom graphs or a suitably defied family of expader graphs) satisfyig these properties Further, the results of this sectio hold regardless of how the fire chooses which eighbours to spread 3 Numerical upper boud From Theorem 4 it follows that ρg, k) teds to as k+ d O the other had, for a give value of d, oe ca fid a better umerical upper boud for ρg, k) At the ed of the process i roud t, there are B t = kt + O) vertices o fire, F t = t protected vertices, ad R t = k + )t + O) eutral vertices that are separated from the fire; that is, there are o edges betwee B t ad R t Suppose ow that for a give x [0, x 0 ] where 0 x 0 is a fixed costat) ad ay y, z [0, ], the expected umber Sx, y, z) of partitios of the vertex set ito three sets B, F, R of kx, x, ad k+)x) vertices i G G,d, respectively, with EB, F ) = y, EF, R) = z, ad EB, R) = 0 is o 3 ) The, the expected umber of cofiguratios with x x 0 is o) ad so aas o such partitio exists by the first momet method This implies that aas the process ca last up to roud x 0 ; that is, aas ρg, k) < kx 0 To fid the optimal value of x 0 we use the pairig model

11 FIGHTING CONSTRAINED FIRES IN GRAPHS For ay x [0, ], it follows that ) ) ) ) kx) kxd xd Sx, y, z) = y)! Md) kx x y y ) ) xd y k + )x)d z)! z z Mkxd y)mxd y z)m k + )x)d z), where Mi) is the umber of perfect matchigs o i vertices; that is, Mi) = i! i/)! i/ To see this, we fix kx vertices kxd poits) to form set B the term kx) ), ad x vertices xd poits) to form set F the term ) kx) x ) Now, we fix y poits i B the term ) kxd ) that correspod to y edges that are icidet to y poits i F the term xd y ) y ) After fixig poits i both B ad F, we eed to coect them i all possible ways the term y)!) Similarly, we geerate all possibilities for edges from F to R the term xd y ) k+)x)d ) z z z)!) Fially, we eed to take a perfect matchig of remaiig poits i B the term Mkxd y)), F the term Mxd y z)), ad R the term M k + )x)d z)) to cosider all possible cofiguratios satisfyig our assumptio Defie ) k gx, y, z, k, d) = kxd ) x k+)xd ) d d/ k+)x) k+)x)d ) y y z z kxd y) kxd y)/ xd y z) xd y z)/ k+)x)d z) k+)x)d z)/ After simplificatio, usig Stirlig s formula that is,! π/e) ), ad takig the expoetial part we obtai that where Sx, y, z) e o) gx, y, z, k, d) = e fx,y,z,k,d)+o), fx, y, z, k, d) = kxd ) l k + k + )xd ) l x + d l d + k + )x) l k + )x) y l y z l z kxd y) lkxd y) xd y z) lxd y z) k + )x)d z) l k + )x)d z) It follows that if max y,z fx, y, z, k, d) < 0, the Sx, y, z) is expoetially small for large) for ay y, z To maximize the fuctio f, we eed to cosider the followig system of partial differetial equatios: 0 = f y = l y + lkxd y) + ldx y z) 0 = f z = l z + ldx y z) + l k + )x)d z) We therefore obtai that y 0 kxd y 0 = z 0 k + )x)d z 0 = dx y 0 z 0,

12 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT which ca be solved umerically Note that y 0 ad z 0 are fuctios of x, as well as of k ad d) Fially, if fx, y 0, z 0, k, d) = fx, y 0 x, k, d), z 0 x, k, d), k, d) = fx, k, d) < 0 for every x [0, x 0 ], we derive that aas ρg, k) < kx 0 We used this approach to obtai asymptotically almost sure upper bouds ud, k) for the survivig rate of radom d-regular graph Whe we quote umerical values for ud, k), we use five decimal places rouded up for upper bouds For compariso purposes, let ūd, k) deote a explicit upper boud followig from Theorem 4 I Figure 3, the values of ud, k) ad ūd, k) are give for k = 3 ad k = 9 ad all d-values betwee 0 ad 00; we also listed the first 8 ad a few more values for higher d i Tables ad The computatios preseted i the paper were performed usig C/C++ The code may be foud o-lie at [6] a) k = 3 b) k = 9 Figure 3 Numerical ad explicit upper bouds for d from 0 to 00 d ud, 3) ūd, 3) d ud, 3) ūd, 3) d ud, 3) ūd, 3) Table Numerical ad explicit upper bouds for k = 3 for some d values 4 k-survivig rates of the ifiite radom graph Firefighter is ofte studied i the cotext of ifiite grid graphs i the plae However, i coutably ifiite graphs, we caot defie the k-survivig rate i the exact same fashio as for fiite graphs For example, we caot divide by the order of a ifiite graph A

13 FIGHTING CONSTRAINED FIRES IN GRAPHS 3 d ud, 9) ūd, 9) d ud, 9) ūd, 9) d ud, 9) ūd, 9) Table Numerical ad explicit upper bouds for k = 9 for some d values atural defiitio is to write the graph as a limit of fiite graphs, ad take the limit of k- survivig rate of graphs i the chai However, differet chais ca lead to differet limitig k-survivig rates Let G t : t N) be a chai of fiite coected graphs; that is, the sets {V G t ) : t N} ad {EG t ) : t N} are well-ordered Defie V G) = t N V G t ), EG) = t N EG t ) We write lim t G t = G, ad say that G is the limit of the chai G t : t N) For example, a ifiite oe way path or ray) is the limit of the chai P t : t N) Let C = G t : t N) be a chai of fiite graphs ote that the limit of C is coected as each G t is) Defie the limitig k-survivig rate of G relative to the chai C) by ρ C G, k) = lim t ρg t, k), assumig this limit exists Observe that ρ C G, k), whe it exists, is a real umber i the iterval [/k + ), ] The value ρ C G, k) may strogly deped o the chai C used, but ot i all cases For example, for a ray P = lim t P t, every chai is a set of paths, ad so ρ C P, k) = The coutably ifiite clique has survivig rate regardless of the chai k+ used each elemet of the chai is a clique) Defie the probability space GN, p) to be graphs with vertex set of positive itegers, ad each distict pair of itegers is joied idepedetly with probability p 0, ) We will call this space the ifiite radom graph Erdős ad Réyi [0] discovered the followig theorem Theorem 8 For p 0, ) with probability, the graph GN, p) is uique up to isomorphism Defie a graph to be existetially closed or ec if all fiite disjoit sets of vertices A ad B oe of which may be empty), there is a vertex z joied to all of A ad to o vertex of B We say that z is correctly joied to A ad B The proof of Theorem 8 follows by provig that with probability, GN, p) is ec, ad the provig that ay two ec graphs are isomorphic The uique isomorphism type of ifiite radom graph is writte R We exploit the followig explicit represetatio of R as a limit graph For a graph H, for each o-empty subset S of V H), add a ew vertex z S joied to S ad to o other vertices The resultig graph, writte H), cotais H as a iduced subgraph Note that if H has order, the the order of H) is + ote that we omit the case with S = as that would itroduce

14 4 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT a isolated vertex) We may iterate this process, so Ht + ) = Ht))) It is easy to see that for a fiite graph H, lim t Ht) is ec, ad so is isomorphic to R Choose t large eough such that Ht) cotais both A ad B A vertex correctly joied to A ad B may be foud i Ht + )) Our mai result i this fial sectio is that for the ifiite radom graph, we ca obtai ay real k-survivig rate i [/k + ), ] Theorem 9 For each real umber r [ k+, ], there is a chai C = G t : t N) of fiite graphs such that ) lim t G t = R ) ρ C R, k) = r We ote that a aalogous result was obtaied for the cop umber of a graph i [5], but with oe importat caveat I the defiitio of so-called cop desity of a coutably ifiite graph relative to a chai of fiite iduced subgraphs that is, the limit of the ratio of the cop umber to the order of the graphs i the chai), graphs i the chai were allowed to be discoected By a result of Frakl [3], if the graphs i the chai are coected, the the limitig cop desity is always 0 he proved that the cop umber of a coected graph of log log log order is O cop desity theorem for R To prove the theorem, we eed the followig two lemmas ) = o)) Hece, Theorem 9 stads i stark cotrast to the aalogous Lemma 0 Fix a fiite graph G of order The, ρg), k) = + o) k + Lemma Fix c a o-egative real umber, a graph G of order N, ad a vertex v of G Defie G) c,v) by first formig G) of order = N + N, the attachig a path of legth c to v The Note that if c 0, the while c implies that ρg) c,v), k) = + cc + ) k+ + o) c + ) + cc + ) k+ c + ) k +, + cc + ) k+ = c + ) k+ + c +c + c +c I particular, Lemma tells us that by choice of c, we ca make ρg) c,v), k) as close as we like to ay fixed real i [ k+, ] Proof of Theorem 9 Fix a sequece c t of o-egative reals such that lim t + c k+ tc t + ) = r c t + ) Note that if r <, the oe ca take a costat sequece I order to get r =, we eed to take ay sequece tedig to ifiity

15 FIGHTING CONSTRAINED FIRES IN GRAPHS 5 Fix G 0 = K For t 0, defie G t+ = G t )) ct,v), where v is chose arbitrarily i G t ) Note that G t : t N) forms a chai C, ad G = lim t G t is ec ad so G = R By Lemma, k+ ρg t+, k) = + c tc t + ) + o) c t + ) Hece, ρ C R, k) = r Proof of Lemma 0 We call vertices i V G))\V G) ew The ew vertices with degree oe are called bad, ad the remaiig ew vertices are good Note that there are bad vertices ad so there are good vertices We cosider cases of where the fire ca break out If a fire breaks out o G, the each vertex of G is joied to ew vertices of degree so the fire ca reach at least oe such a vertex i the ext roud Spreadig a fire from this vertex back to G, the firefighter ca oly save at most + o)) of these Now a fire spreads to ew vertices Sice all but k+ +o))/k+) = o ) ew vertices are adjacet to the fire, the firefighter ca oly save at most +o)) ew vertices If a fire starts at a good ew vertex, the at least oe vertex k+ of G ca be set o fire i the ext roud Hece, i the remaiig rouds we are back i the first case If a fire breaks out o a bad vertex, the all but oe vertex of G) ca be saved We therefore have that ρg), k) + + o)) + k+ ) ) = + o) k + The proof of the lemma ow follows from ) Proof of Lemma By Lemma 0, ρg), k) = +o) If a fire breaks out o G), the the k+ firefighter chooses to save v i the first roud or save the vertex o the path adjacet to v i the case v is o fire) The + o)) vertices ca be saved i G by protectig ay vertex k+ i G) regardless of the fire s strategy As the fire caot spread to the path, a additioal + o))c vertices are saved Note that this strategy is asymptotically the best possible; if the firefighter decides ot to save v i the first roud ad focus o protectig graph G), he would ot be able to save more tha + o)) vertices i G) by Lemma 0 k+ If the fire breaks out o a vertex of the path ot i G), the the firefighter ca cotai it to save all but at most two vertices i G) c,v) Hece, ρg) c,v), k) = + o)) + c) + cc + ) k+ c + ) k+ = + o)) + c) + c + c) c + ) = + cc + ) k+ + o) c + ) 5 Ackowledgemets We thak the aoymous referees whose commets ad correctios greatly improved the paper

16 6 ANTHONY BONATO, MARGARET-ELLEN MESSINGER, AND PAWE L PRA LAT Refereces [] N Alo, FRK Chug, Explicit costructio of liear sized tolerat etworks, Discrete Math, 7 988) 5 9 [] N Alo, VD Milma, λ, isoperimetric iequalities for graphs ad supercocetrators, J Combiatorial Theory, Ser B ) [3] N Alo, JH Specer, The Probabilistic Method, Wiley, 99 Secod Editio, 000) [4] B Bollobás, A probabilistic proof of a asymptotic formula for the umber of labelled regular graphs, Europea Joural of Combiatorics 980) 3 36 [5] A Boato, G Hah, C Wag, The cop desity of a graph, Cotributios to Discrete Mathematics 007) [6] C/C++ code Accessed October 5, 0 [7] L Cai, W Wag, The survivig rate of a graph for the firefighter problem, SIAM Joural of Discrete Mathematics, 3 009) [8] L Cai, Y Cheg, E Verbi, Y Zhou, Survivig rates of graphs with bouded treewidth for the firefighter problem, SIAM Joural of Discrete Mathematics 4 00) [9] M Devli, S Hartke, Fire cotaimet i grids of dimesio three ad higher, Discrete Applied Mathematics ) [0] P Erdős, A Réyi, Asymmetric graphs, Acta Mathematica Academiae Scietiarum Hugaricae 4 963) [] S Fibow, P Wag, W Wag, O the survivig rate of a ifected etwork, Theoretical Computer Sciece 4 00) [] S Fibow, G MacGillivray, The firefighter problem: a survey of results, directios ad questios, Australasia Joural of Combiatorics ) [3] P Frakl, O a pursuit game o Cayley graphs, Combiatorica 7 987) [4] J Friedma, A proof of Alo s secod eigevalue cojecture, Memoirs of the America Mathematical Society 95, 008 [5] B Hartell, Firefighter! A applicatio of domiatio Presetatio at the 5th Maitoba Coferece o Combiatorial Mathematics ad Computig, Uiversity of Maitoba, Wiipeg, Caada, 995 [6] B Hartell, Q Li, Firefightig o trees: How bad is the greedy algorithm?, I: Proceedigs of the Thirty-first Southeaster Iteratioal Coferece o Combiatorics, Graph Theory ad Computig, 000 [7] S Hoory, N Liial, A Wigderso, Expader graphs ad their applicatios Bull Amer Math Soc ) [8] A Kig, G MacGillivray, The firefighter problem for cubic graphs, Discrete Math 30 00) 64 6 [9] P Pra lat, Graphs with average degree smaller tha 30 are burig slowly, preprit [0] P Pra lat, Sparse graphs are ot flammable, preprit [] NC Wormald, Models of radom regular graphs Surveys i Combiatorics, 999, JD Lamb ad DA Preece, eds Lodo Mathematical Society Lecture Note Series, vol 76, pp Cambridge Uiversity Press, Cambridge, 999 Departmet of Mathematics, Ryerso Uiversity, Toroto, ON, Caada, M5B K3 address: aboato@ryersoca Departmet of Mathematics & Computer Sciece, Mout Alliso Uiversity, Sackville, NB, Caada, E4L E6 address: mmessiger@mtaca Departmet of Mathematics, Ryerso Uiversity, Toroto, ON, Caada, M5B K3 address: pralat@ryersoca