Largest and smallest minimal percolating sets in trees

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1 Lagest and smallest minimal pecolating sets in tees Eic Riedl Havad Univesity Depatment of Mathematics Submitted: Sep 2, 2010; Accepted: Ma 21, 2012; Published: Ma 31, 2012 Abstact Oiginally intoduced by Chalupa, Leath and Reich fo use in modeling disodeed magnetic systems, -bootstap pecolation is the following deteministic pocess on a gaph. Given an initial infected set, vetices with at least infected neighbos ae infected until no new vetices can be infected. A set pecolates if it infects all the vetices of the gaph, and a pecolating set is minimal if no pope subset pecolates. We conside minimal pecolating sets in finite tees. We show that if A is a minimal pecolating set on a tee T with n vetices and l vetices of degee less than (leaves in the case = 2), then ( 1)n+1 A n+l +1. Moeove, we show that the diffeence between the sizes of a lagest and smallest minimal pecolating sets is at most ( 1)(n 1). Finally, we descibe O(n) algoithms fo 2 computing the lagest (fo = 2) and smallest (fo 2) minimal pecolating sets. 1 Intoduction In this pape we study the following deteministic pocess on a gaph G, known as - neighbo bootstap pecolation. Let A 0 = A be a subset of the vetices of G, and fo each t 1, let A t = A t 1 B t, whee B t is the set of vetices with at least neighbos in A t 1. We wite A = t=0a t, and say that A pecolates if A = V (G). We think of the pocess as modeling the spead of infection, with A the set of initially infected vetices and A the set of eventually infected vetices. A set A is a minimal pecolating set if A pecolates but no pope subset of A pecolates. Note that such sets must exist, as G = G. It tuns out that minimal pecolating sets ae not necessaily all the same size. In this pape, we concen ouselves with the following questions: given a cetain finite gaph, what ae the sizes of the lagest and smallest minimal pecolating sets? What bounds can we pove on the sizes of these sets? Can we povide examples of such sets? This pape consides these questions fo tees. Bootstap pecolation was intoduced in 1979 by Chalupa, Leath, and Reich in [9] who wee motivated by applications to disodeed magnetic systems. It can be used to the electonic jounal of combinatoics 19 (2012), #P64 1

2 model many diffeent eal-wold phenomena, including magnetic mateicals, fluid flow in ocks, and compute stoage systems. Fo moe details on applications of bootstap pecolation, see the suvey aticle by Adle and Lev [1]. Theoetical wok on -bootstap pecolation has focused mostly on vaiants of the following pobabilistic question. Given a gaph G, choose a andomly infected set of of vetices A by letting each v G be in A independently with pobability p. Then what is p c (G, ) = inf{p P(Apecolates) 1/2}? Aizenman and Lebowitz and Cef and Ciillo did foundational wok fo the poblem on gids [n] d in [2, 7], and Cef and Manzo [8] poved that ( ) d +1 p c ([n] d 1, ) = Θ, log ( 1) n whee log () (x) is log(log( log(x))) ( times). Holoyd [10] found moe pecise estimates fo = 2, d = 2 and Balogh, Bollobás, Duminil-Copin and Mois [3, 4] found pecise asymptotics fo all and d. Balogh, Bollobás and Mois also found asymptotic esults fo d log(n), and n and d both inceasing [5]. Balogh, Pees and Pete found p c fo infinite tees in [6]. Wite m(g, ) fo the size of a smallest minimal pecolating set, and E(G, ) fo the size of a lagest minimal pecolating set unde -bootstap pecolation. In this pape we bound m and E. As seen in many cases (such as [11]), infomation about the size and stuctue of minimal pecolating sets can often be both useful in answeing the oiginal pobabilistic pecolation questions and inteesting in tems of gaining insight into the pecolation pocess. Minimal pecolating sets ae cucial to undestanding the stuctue of pecolating sets, as any pecolating set necessaily contains a minimal pecolating set. Bounding the maximum and minimum sizes of minimal pecolating sets is a basic fist step in investigating minimal pecolating sets. Moeove, studying E and m gives a sense of how badly the geedy algoithm could fail in minimizing the numbe of sites equied fo pecolation. Less wok has been done on the extemal poblem than on the pobabilistic poblem. It is a folk-loe fact that fo = 2, the smallest minimal pecolating set in the n n gid has size n. Pete [12] computes m([n] d, ) up to a constant facto fo evey fixed d and, and poves moe pecise estimates in special cases. Mois [11], answes a question of Bollobás and finds asymptotic bounds fo the size of a lagest minimal pecolating set in [n] 2, showing that it lies between 4n2 n2 and. We investigate E and m fo finite tees Since tees ae a divese family of gaphs, it is difficult to find a simple fomula fo E and m. Indeed, thee is no known simple numeical chaacteization of tees which can be used to detemine the sizes of thei minimal pecolating sets. Candidates such as numbe of vetices o degee sequence simply do not cay enough infomation. Fo example, the gaphs in Figue 1 have the same numbe of vetices and the same degee sequence but diffeent smallest minimal pecolating set sizes. Ou main esults ae bounds on the sizes of minimal pecolating sets and O(n) algoithms fo computing the sizes of minimal pecolating sets. Moe specifically, in Section 2 we use an edge-counting technique to obtain thee bounds on sizes of minimal pecolating sets, which when combined give the following. the electonic jounal of combinatoics 19 (2012), #P64 2

3 Figue 1: Smallest minimal pecolating sets in two tees with the same numbes of vetices and degee sequences but diffeent sizes of the smallest minimal pecolating set. Main Theoem. Let T be a tee with n vetices and l leaves. Then and ( 1)n + 1 m(t, ) E(T, ) n + l + 1 E(T, ) m(t, ) ( 1)(n 1) 2. We exhibit infinite families fo which all of the bounds ae shap except fo the bound on E(T, ) m(t, ) with = 2; in this case we exhibit an infinite family of gaphs fo which E(T, ) m(t, ) is within 1 of the bound. We pove ou main theoem in thee pats. Poposition 3 is the lowe bound on m(t, ), Theoem 4 is the uppe bound on E(T, ) and Theoem 5 is the bound on E(T, ) m(t, ). In Section 3 we descibe O(n) algoithms fo computing m(t, ) fo all 2 and E(T, 2). 2 Bounds In this section we give some bounds on the maximum and minimum size of a minimal pecolating set, as well as the diffeence between them. Thoughout we let T = n unless othewise stated. We give thee main bounds. Fist, we show m(t, ) ( 1)n+1. Note that this is shap fo a complete -ay tee. Second, if l is the numbe of vetices of degee less than, we show E(T, ) n+l ( 1)(n 1). Finally, we show E(T, ) m(t, ) In ode to pove ou esults, we need to define the notion of a wasted edge. Let G be a gaph and fix a pecolating set A on G. Intuitively, some of the edges of G ae used, and ae necessay fo A to pecolate, wheeas othes ae wasted, because we can emove them and the gaph will still pecolate. We now make this intuitive notion pecise. We give each edge of ou gaph one of two designations: wasted o used, and we give each used edge a diection. Stat with a set A of initially infected vetices. Each time a new vetex v is infected in the pecolation pocess, choose of the edges connecting v to infected vetices and call them used, diecting the electonic jounal of combinatoics 19 (2012), #P64 3

4 Figue 2: A choice of wasted edges in a boom. them towad v. Continue this pocess until all of the vetices ae infected. Call all of the undiected edges wasted. Note that it is impossible to give an edge moe than one diection, as diections ae only given to edges incident to uninfected vetices, and once an edge is given a diection, both of the vetices it is incident to ae infected. If G is a gaph and A a pecolating set, then given a choice of wasted edges, A will pecolate in G minus the wasted edges. See Figue 2 fo an example of a choice of wasted edges in a boom. Let w be the numbe of wasted edges. Now, in most cases thee is feedom involved in choosing wasted and used edges, and it is not immediately clea that w is well-defined. The next poposition shows that it is. Poposition 1. Let G be a gaph with e(g) edges and let A be a pecolating set in G. Then w, the numbe of wasted edges, is well defined fo any choice of used edges, and w = e(g) ( G A ). Poof. Each vetex v of G \ A is infected by exactly used edges diected towad v. Since each edge is given only one diection, we know that thee ae exactly ( G A ) used edges. Hence, thee ae e(g) ( G A ) wasted edges. Coollay 2. If T is a tee with T = n, and w is the numbe of edges wasted fo a pecolating set A, then w = ( 1)n 1 + A. As an immediate application of the notion of wasted edges, we deduce a lowe bound on m(t, ) fo tees. Poposition 3. Let T be a tee, T = n > 2. Then m(t, ) ( 1)n + 1 with equality if and only if thee exists a minimal pecolating set with no wasted edges. the electonic jounal of combinatoics 19 (2012), #P64 4

5 Note that this bound is shap. Fo example, if = 2, paths with an odd numbe of vetices and complete binay tees both have m(t ) = n+1. Fo > 2, complete ( 1)-ay 2 and -ay tees have m(t ) = ( 1)n+1. Indeed, fo complete -ay tees, m(t ) = E(T ), so this also gives a shap lowe bound fo E(T, ). Poof. The numbe of wasted edges is clealy nonnegative: w = ( 1)n 1 + A 0. Thus, which gives A ( 1)n + 1 A ( 1)n + 1. Note that while we ae only concened with tees in this pape, the idea of the above poof of Poposition 3 woks fo any gaph G, giving m(g, ) G e(g). We now tun to uppe bounds on the size of minimal pecolating sets. Fo > 2, paths ae an example of tees fo which E(T, ) = m(t, ) = T. Fo = 2, we have tivially that E(T, 2) T 1 fo T > 2, as any tee of size at least 3 must have a vetex of degee at least 2, so it is impossible fo V (G) to be a minimal pecolating set. Howeve, fo stas we have E(T, 2) = m(t, 2) = T 1. Hence, the shap uppe bounds which hold fo all tees ae not paticulaly inteesting. Note, howeve, that these examples all have lots of vetices of degee less than, which foced the sizes of minimal pecolating sets to be lage. If we keep tack of how many such vetices we have, we can get a moe inteesting bound. Theoem 4. Let G be a gaph with l vetices of degee less than, G = n. Then E(G, ) n + l + 1. Poof. The key obsevation is that if A is a minimal pecolating set, and v A of degee at least, then v must be incident with at least one used edge. Since a used edge is incident with at most one vetex in A, this means that the numbe of used edges is at least the numbe of vetices in A of degee geate than o equal to. Thus, (n A ) A l o A n + l + 1. the electonic jounal of combinatoics 19 (2012), #P64 5

6 Figue 3: Example of gaph with maximum E(T,4) m(t,4) n 1. The bound in Theoem 4 is shap (up to intege pats) fo paths, stas, and booms (paths with some numbe of pendant edges attached to one of the leaves) fo = 2. Fo 3, the esult is shap fo paths, stas (up to intege pats) and fo gaphs obtained by attaching 1 pendant edges to each leaf in a sta with leaves. We have just given bounds on both m(t, ) and E(T, ) sepaately. We now bound thei diffeence. Since the possible diffeence in sizes will obviously gow with T, we conside the lagest possible diffeence as a function of n = T. Theoem 5. Fo any tee T with T = n > 1 we have E(T, ) m(t, ) ( 1)(n 1) 2. Befoe embaking on the poof of Theoem 5, we give some examples to show that the above bound is shap. Constuct the following tee T. Stat with a vetex c. Then attach vetices to c. Finally, attach 1 vetices to each new vetex. Fo = 2 this is simply P 5. Now, fo these tees, we have E(T, ) = 2 (given by the set T \ c) and m(t, ) = ( 1) + 1 (given by the set consisting of c and the leaves of T ). Thus, E(T,) m(t,) = 1. See Figue 3. T 1 2 The next natual question to ask is whethe thee is an infinite family of gaphs that have E(T,) m(t,) = 1. Fo > 2, the answe to this question is yes. We constuct ou T 1 2 family inductively. Suppose T 1 and T 2 ae two tees with maximal E(T i, ) m(t i, ). Let w 1 be a leaf of T 1 and let w 2 be a leaf of T 2. Then the tee obtained by identifying w 1 and w 2 into one vetex (of degee 2) will be anothe tee which satisfies the equality. Fo = 2, the above constuction will not wok. Fo > 2, vetices of degee 2 behave like leaves in the sense that they must be infected in any minimal pecolating set, but fo = 2, this is cetainly not the case. We do not know of an infinite family of gaphs which satisfy the equality fo = 2. Howeve, we can display an infinite family of the electonic jounal of combinatoics 19 (2012), #P64 6

7 Figue 4: Example of 6 P 5 s joined togethe at thei leaves. gaphs which get abitaily close to the (pehaps moe natual) bound E(T,2) m(t,2) < 1. n 4 Take k P 5 s and choose a leaf of each P 5. Glue all of the P 5 s togehte by identifying these leaves into a single vetex v. Then the esulting gaph will have E(T, 2) = 3k (the minimal pecolating set consists of all leaves, neighbos of leaves and neighbos of v) and m(t, 2) = 2k + 1 (the minimal pecolating set consists of v, all leaves, and all vetices of distance 2 fom v). Thus, fo this tee E(T,2) m(t,2) = k 1, which gets abitaily close to n 4k+1 1 as k gets lage. 4 In poving Theoem 5, we fist show the esult fo a lage class of tees, then extend the esult to all tees. Befoe poceeding, we intoduce some teminology. Recall that A is the set of vetices eventually infected by an initial set A. In case the ambient gaph is not obvious, we wite A G fo the set of vetices in G eventually infected by A unde the pecolation pocess. A vetex v is said to be taceable back to leaves if v L, whee L is the set of leaves. Note that leaves ae tivially taceable back to leaves. We now pove a cucial lemma. Lemma 6. Let T be a tee with T > 2 such that no nonleaf is taceable back to leaves, and evey nonleaf has degee at least. Let A be a minimal pecolating set. Then a faction of at most 1 of the edges of T ae wasted. That is, w (n 1)( 1). Poof. Fist, we show that we can choose a set of wasted edges such that evey wasted edge is incident with a nonleaf in A. Suppose we ae given a choice of wasted edges such that k wasted edges that ae incident with a nonleaf in A, with k stictly less than the total numbe of wasted edges. We constuct a choice of wasted edges such that k + 1 wasted edges ae incident with a nonleaf in A. Let e be a wasted edge that is not incident to a nonleaf in A. Since n > 2, e will be incident with at least one nonleaf, so let v be a nonleaf incident with e. Let Ω be the set the electonic jounal of combinatoics 19 (2012), #P64 7

8 of diected paths of used edges fom nonleaf vetices in A to v. Since v is not taceable back to leaves, Ω is non-empty, so choose a path (v 0,, v d ) Ω, with v 0 a nonleaf in A, v d = v. We now exhibit a choice of wasted edges with k + 1 wasted edges incident with a nonleaf in A: take the oiginal choice of wasted edges, except designate e as used and diect it towad v, evese the diection of evey edge v i v i+1 fo 1 i d 1 and designate v 0 v 1 as wasted. This will be a legal choice of wasted and used edges, and since v 0 is a nonleaf in A, we will have exactly one moe wasted edge which is incident with a nonleaf in A. Now, any vetex v A is adjacent to at most 1 wasted edges, since if v wee adjacent to o moe, A \ v would pecolate, which contadicts the minimality of A. Thus, if B A is the set of nonleaves, then w ( 1) B. Let u be the numbe of used edges. Then evey vetex of B is incident with at least one used edge, and since edges between two elements of A cannot be used, we have Thus w ( 1)u. Now, since u+w n 1 u B. = 1, we have w (n 1)( 1). Now we use the above esult to pove an uppe bound fo E(T, ) fo the specific class of tees to which we have esticted ouselves. Poposition 7. Let T be a tee with no vetices taceable back to leaves and with the popety that evey nonleaf has degee at least. Then E(T, ) (2 1)n Poof. Let A be a minimal pecolating set with A = E(T, ). Fom Lemma 6, we have Using Poposition 1 we have w (n 1)( 1). ( 1)n + 2 A n n + 1 A (2 1)n Coollay 8. Fo tees T with evey nonleaf having degee at least and no nonleaf vetices taceable back to leaves, E(T, ) m(t, ) ( 1)(n 1) 2. the electonic jounal of combinatoics 19 (2012), #P64 8

9 Poof. Simply subtact the inequalities fom Poposition 3 and Poposition 7. Poof of Theoem 5. Now, we finally extend ou esult to geneal tees T using Coollay 8. Fist, we dispose of the condition that no nonleaves ae taceable back to leaves. Suppose, to get a contadiction, that T is a smallest tee with E(T, ) m(t, ) > ( 1)( T 1) such that evey nonleaf of T has degee at least. Then (since by Coollay 2 8 any such tee T must have a vetex taceable back to leaves) thee is a vetex v T with at least pendant edges. Let L be the set of leaves attached to v. Let T 1,..., T k be the connected components of T \ (v L). Let T i be T i with one leaf w i adjoined to the unique neighbo of v in T i. Note that all nonleaves in T i have degee at least. Now, thee is a canonical bijection between minimal pecolating sets on the disjoint union of the T i s and minimal pecolating sets of T. Take a minimal pecolating set A of T. Note that v / A. Fo each i, set A i = (A T i ) w i. Then each A i will be a minimal pecolating set of T i, and similaly, given A i which ae minimal pecolating sets in the T i, L i (A i \ w i ) will be a minimal pecolating set in T. Thus, we have T = i ( T i 1) L, while E(T, ) m(t, ) = i (E(T i, ) m(t i, )). Hence, E(T, ) m(t, ) T 1 i = (E(T i, ) m(t i, )) i i ( T < (E(T i, ) m(t i, )) i 1) + L i ( T i 1) { } E(Ti, ) m(t i, ) max i T i 1 Thus, one of the T i also satisfies E(T i, ) m(t i, ) > ( 1)( T i 1). Howeve, this contadicts minimality of T, as T i < T. Thus, ou esult holds fo all tees in which evey 2 nonleaf has degee at least. Now we dispose of the condition that evey nonleaf of T has degee at least, so this paagaph applies only fo > 2. To get a contadiction, find a smallest tee T such that ( 1)( T 1) E(T, ) m(t, ) >. By the esult of the pevious paagaph, we can find 2 a vetex v of T of degee 1 < d <. Thus, v will have to be infected in any minimal pecolating set. Let T 1,..., T n be the connected components of T \ v with an exta leaf w i added to the vetex connected to v. As above, thee is a canonical bijection between minimal pecolating sets in the disjoint union of the T i s and minimal pecolating sets of T. Namely, given a minimal pecolating set A of T, (A T i ) w i will be a minimal pecolating set fo each of the T i, and if A i ae minimal pecolating sets fo each of the T i, v i (A i \ w i ) will be a minimal pecolating set of T. Now, conside the quantity E(T,) m(t,). We have E(T, ) m(t, ) = T 1 i (E(T i, ) m(t i, )). Moeove, T = i ( T i 1) + 1. Thus, we have E(T, ) m(t, ) i = (E(T { } i, ) m(t i, )) E(Ti, ) m(t i, ) T 1 i ( T max. i 1) i T i 1 Thus, one of the T i also satisfies E(T i, ) m(t i, ) > ( 1)( T i 1). This contadicts the 2 minimality of T. Hence, ou esult holds fo all tees. the electonic jounal of combinatoics 19 (2012), #P64 9

10 3 Algoithms In this section, we descibe algoithms to compute m(t, ) and E(T, 2). The basic idea of the algoithms is to ecusively solve the poblem by finding and modifying cetain specific subtees. Note that in geneal, extemal (that is, lagest and smallest) minimal pecolating sets ae not unique, and if we un ou algoithm in a slightly diffeent ode we could end up with a diffeent extemal minimal pecolating set, although the size will of couse be the same. Befoe pesenting the algoithms, we need to define a few tems. We say that H is a tailing path (o a tailing P k ) if H is a path of length k 1 and is connected to G \ H by a single edge going fom one of the ends of the path to the est of the gaph. We say that H is a tailing sta if H is a K 1,l with H connected to G \ H by a single edge fom the cente vetex of the sta H. We will also need the notion of a pseudo-sta. We say that a tee H with cente vetex v is a pseudo-sta with cente v if evey vetex of H has distance at most 2 fom v. Define a tailing pseudo-sta to be a subtee that is a pseudo-sta connected to the est of G by a single edge fom the cente v of the pseudo-sta. Note that stas ae pseudo-stas, and tailing stas ae tailing pseudo-stas. A staight pseudo-sta is a pseudo-sta fo which evey vetex except the cente has degee at most 2, while a banched pseudo-sta is any pseudo-sta which is not staight. We descibe two diffeent algoithms, mset and ESet, fo computing m(t, ) and E(T, ) espectively. Both of the algoithms pesented consist of the same basic steps. Note Step 1 is unnecessay in the case whee = 2, and hence is unnecessay fo ou algoithm ESet. Step 1 Repeatedly pefom the eduction pocedue (to be descibed) until evey vetex has degee 1 o at least. Step 2 Identify a tailing sta o pseudo-sta that can be educed (we will descibe below which tailing subgaphs can be educed in which instances). Step 3 Fom the gaph T by modifying the tailing sta, pseudo-sta o path (in a manne that will be descibed below). Step 4 Set A = mset(t, ) o A = ESet(T, ). Step 5 Modify A (in a manne that will be descibed below) to obtain A, a smallest o lagest minimal pecolating set of T. Step 6 Output A. Steps 4 and 6 ae self-explanatoy. Steps 2, 3 and 5 depend on the specifics of and whethe o not we ae computing m(t, ) o E(T, ), and we elaboate on them in detail below. The pupose of Step 1 is to ensue that evey non-leaf has degee at least, and it is simple enough that we descibe it now. the electonic jounal of combinatoics 19 (2012), #P64 10

11 Step 1 is pefomed as follows. Iteate though each vetex v of degee less than. Let T i be the connected components of T \ v. Let T i be T i w i, whee w i is a single leaf attached to the unique neighbo of v in T i. Pefom the algoithm on all of the T i, obtaining minimal pecolating sets A i w i. Let A = v A i be the output of the algoithm. 3.1 m(t, ) Because the poofs of the algoithms ae often long and a bit tedious, we sketch many of them. We stat by elaboating on step 2. When computing m(t, ), we educe by identifying tailing stas, so in ode to implement step 2, we need to know that we can always identify tailing stas, and pove that we can do so efficiently. Lemma 9. Evey tee has a tailing sta. Poof. Find a longest path in T, with v the second-to-last vetex of the path. Then v will be the cente of a tailing sta. The above poof suggests an algoithm fo identifying tailing stas, and this is essentially the method we adopt. Howeve, because the poof of O(n) untime is somewhat inticate, we defe discussion of this until afte descibing Steps 3 and 5. Steps 3 and 5 divide into the following two cases, depending on the numbe of leaves of the tailing sta. Let v be the cente vetex of the tailing sta identified in Step 2, and L the set of leaves of the tailing sta. Case A: L <. Note that because of Step 1, this implies L = 1. In the case of = 2, this is simply a tailing P 2. When pefoming Step 3 in this case, we set T = T \ (v L). When pefoming Step 5 in this case, we set A = A L. Case B: L. When pefoming Step 3 in this case, we set T = T \ L. When pefoming Step 5 in this case, we set A = A L \ v. Note that v is a leaf in T, so v will always be in A. This gives a complete desciption of how to pefom ou algoithm fo m(t, ). Because the poof that ou algoithm woks is tedious and elatively intuitive, we meely sketch the poofs. We stat with Case B. Lemma 10. Given a gaph T with a tailing sta v L (v the cente, L the leaves) with L, and given a smallest minimal pecolating set A fo T = T \ L, we then have A L \ v is a smallest minimal pecolating set fo T. Poof. In any minimal pecolating set of T, evey vetex of L must be infected. The vetices of L alone ae sufficient to infect v. Thus, when constucting a minimal pecolating set of T, we can imagine that v is a leaf and that the vetices of L do not exist. The poof fo Case A equies one auxiliay lemma. the electonic jounal of combinatoics 19 (2012), #P64 11

12 Lemma 11. If T is a tee and w is a leaf, then m(t \ w, ) m(t, ) m(t \ w, ) + 1. Poof. The fist inequality holds because given a minimal pecolating set A of T, we can constuct a pecolating set A of T \ w of size A by taking A = A v \ w, whee v is the unique neighbo of w. The second inequality holds because given a minimal pecolating set A of T \ v, A v will be a pecolating set of T. Now we pove ou algoithm woks fo Case A. Lemma 12. If T is a tee with tailing sta v L (v the cente, L the set of leaves) and L = 1, and if A is a smallest minimal pecolating set fo T = T \ (v L), then A L is a smallest minimal pecolating set fo T. Poof. We see immediately tha A L must pecolate and that L must be in any minimal pecolating set, so ou only concen is that thee might be some minimal pecolating set A of T which contains v and has size stictly smalle than A + L. By Lemma 11 this is impossible, since A \ L will be a minimal pecolating set of T \ L. Thus, we have shown that ou algoithm does indeed poduce a smallest minimal pecolating set. Moeove, it is clea that each step educes the numbe of edges by at least one, and the time taken at each step is linea in the numbe of edges emoved, so the time spent on each step is linea in the numbe of edges. It emains to sketch how to quickly identify tailing stas. Poposition 13. It is possible to efficiently identify tailing stas so that the algoithm mset uns in O(n) time. Poof. Note that the total time taken to pefom Step 1 is linea in the numbe of edges, so the total time spent on Step 1 is O(n). Note also that Step 1 peseves the numbe of edges, so epeatedly applying Step 1 will not cause the poblem to gow too much. At the vey beginning of the algoithm (befoe doing any ecusion), make a list VetList of all of the non-leaves, and fo each vetex in VetList, make a list of its non-leaf neighbos. This entie pocess gows linealy with the numbe of edges of the tee, and so will take O(n) time. Note that a tailing sta can be chaacteized as a non-leaf with exactly one non-leaf neighbo. Let CuentVet be the fist element of VetList. Check if CuentVet has only one non-leaf neighbo. If not, continue and set CuentVet to be the next element in VetList. If so, we have identified a tailing sta. Set CuentVet to the next element of VetList, pefom Steps 3-5 and update VetList and the lists of non-leaf neighbos. Now, in foming T, we emove pecisely one non-leaf neigbo fom a vetex CheckVet. Check to see if CheckVet is the cente of a tailing sta. If so, educe, update VetList and the lists of non-leaf neighbos, and find the new CheckVet. If not, continue with VetList, checking to see if CuentVet is a tailing sta. Afte we have gone completely though VetList, we will be done. the electonic jounal of combinatoics 19 (2012), #P64 12

13 3.2 E(T, ) In this section, we only conside the case = 2. As in ou algoithm fo m(t, ), we begin by poving that we may always apply Step 2. Howeve, in this case the thee types of tailing subgaphs ae tailing stas, tailing P 3 s and tailing staight pseudo-stas. Lemma 14. Evey tee contains a tailing sta, a tailing P 3 o a tailing staight pseudosta. Poof. Let v be the thid-to-last vetex in a longest path in T. If the degee of v is 2, then eithe v is pat of a tailing P 3 o one of the neighbos of v is the cente of a tailing sta. If the degee of v is geate than 2, then v is the cente of a tailing pseudo-sta. If the tailing pseudo-sta is staight, then we ae done. If it is banched, then one of the neighbos of v will be a tailing sta. The poof that we can always efficiently identify tailing stas and psuedo-stas is simila enough to Poposition 13 that we omit it, although thee ae a few exta complications (we need to look fo vetices that have only one non-leaf neighbo and that ae not pat of a tailing P 2 ). Now we elaboate on how to pefom Steps 3 and 5. Thee ae fou diffeent cases. Case A: Tailing sta v L, with v the cente vetex and L the set of leaves. When pefoming Step 3 in this case, let T = T \ L. When pefoming Step 5 in this case, let A = A L \ v. Case B: Tailing staight pseudo-sta with at least 2 leaves attached diectly to the cente vetex, with v the cente vetex of the pseudo-sta, S the est of the vetices of the pseudo-sta, and L S the set of leaves of the pseudo-sta. When pefoming Step 3 in this case, let T = T \S. When pefoming Step 5 in this case, let A = A L\v. Case C: Tailing P 3, with u the leaf, v the unique neighbo of u and w the thid vetex of the pseudo-sta. When pefoming Step 3, let T = T \ {u, v, w}. When pefoming Step 5, let A = A {u, v}. Case D: Tailing staight pseudo-sta with at most one leaf attached diectly to the cente vetex, with v the cente vetex, S the est of the pseudo-sta, L S the set of leaves of the pseudo-sta. When pefoming Step 3 in this case, let T = T u\s, whee u is a leaf attached diectly to v. When pefoming step 5, thee ae two cases. If v A, then let A = A X \ {u, v}, whee X S is a set containing L which contains pecisely two neighbos of v. (Thus, X will be eithe L union one non-leaf o it will be L union two non-leaves.) If v / A, then let A = A Y \ u, whee Y S is a set containing L which contains pecisely one neighbo of v. (Thus, Y will be eithe L, o it will be L union one non-leaf.) Note that in geneal thee will be many possible choices fo X and Y, and diffeent choices will lead to diffeent lagest minimal pecolating sets (although all such sets will of couse have the same size). the electonic jounal of combinatoics 19 (2012), #P64 13

14 The poofs of Cases A and B ae essentially the same as the poof of Case A in the mset lemma, and so we omit them. Thus, it emains to pove the coectness of Cases C and D. We need the analogue of Lemma 11 (we only equie the second inequality, but we pove both fo completeness, and because the poof of the fist is easie). We need some teminology fo this Lemma. We say that a subtee T T is intenally spanned by a set A T if A T T = T. Lemma 15. Let T be a tee and v a leaf. Then E(T \ v, 2) E(T, 2) E(T \ v, 2) + 1. Poof. The poof of both inequalities equies a caeful analysis of what U = T \ A \ w looks like, whee A is a minimal pecolating set and w A is any vetex. It is not had to see that U will be a connected subgaph of T containing w, that is, U is a subtee containing w. Moeove, it is also not too had to see that since T is a tee (and hence thee is a unique path between any two vetices), evey vetex in U except possibly fo w will have pecisely 1 neighbo in A \ w. Convesely, it is clea that if A is a minimal pecolating set and S A is a nonempty subset satisfying U = T \ A \ S is a subtee such that evey vetex in U has pecisely 1 neighbo in T \ U, then S = 1 (because if we add a vetex of U to A we will infect all of T ). We now tun to the fist inequality. Let A T \ v be a lagest minimal pecolating set. Then if A v is a minimal pecolating set in T, we ae done, so suppose not. Let X A be a set of vetices such that A v \ X is a minimal pecolating set of A. We see that U = T \ A \ X T is a subtee of T such that evey vetex of U except fo v has pecisely 1 neighbo in \ A \ X. Thus, U \ v = A \ X T is a subtee of T such that evey vetex has degee 1. Since A is minimal, this shows X = 1, which poves the fist inequality. Now let us conside the second inequality. Let T be a minimal counteexample, i.e. a smallest tee such that E(T, 2) > E(T \ v, 2) + 1. Let A be a lagest minimal pecolating set of T. Let w be the unique neighbo of v. If w A, then we see that A \ v is a lagest minimal pecolating set of T \ v which would pove the esult, so suppose w / A. If A w \ v wee minimal, we would be done, so we may suppose that A w \ v is not minimal. Let T 1,, T k be the connected components of T \ {v, w}, and let A i = A T i. Then we see that pecisely one of the T k, say T 1, is intenally spanned (thee must be at least one since A pecolates, and thee must be at most one since A w \ v is not minimal). By applying the lemma to w T 1, we may find a set A 1 T 1 such that w A 1 is a minimal pecolating set of w T 1 of size at least A 1. Then w A 1 k i=2 A i is a minimal pecolating set of T \ v of size at least A 1. Note that while the fist inequality can be poved in exactly the same fashion fo geneal, the second inequality is false fo geneal. Fo example, conside the following tee T. Stat with a vetex c, adjoin 1 neighbos to it, then adjoin 1 leaves to each neighbo of c. The vetex c will be infected in any pecolating set, and c and the leaves will infect all of T, so E(T, ) = ( 1) = Howeve, if we adjoin a single additional leaf v to c, then we have a minimal pecolating set of T v given by all of the leaves and all of the neighbos of c. Thus, E(T v, ) = ( 1) + 1 = the electonic jounal of combinatoics 19 (2012), #P64 14

15 The failue of this lemma fo geneal pevents us fom easily extending ou algoithm fo E(T, 2) to geneal. In fact, without this lemma (o possibly a much moe lengthy analysis of vaious tailing subgaphs), it is difficult to imagine what a linea-time algoithm fo computing E(T, ) would look like. Using Lemma 15 we can pove the coectness of ou algoithm fo Case C. Lemma 16. If {u, v, w} T is a tailing P 2 with u the leaf, v the unique neighbo of u, then any set of the fom {u, v} A, with A a lagest minimal pecolating set of T \ {u, v, w}, is a lagest minimal pecolating set of T. Poof. It is clea that any set of the fom {u, v} A will be a minimal pecolating set of T, and it is also clea that all such sets will have size E(T \ {u, v, w}, 2) + 2. Thus, it emains to show E(T, 2) E(T \ {u, v, w}, 2) + 2. Now, it is clea that fo any minimal pecolating set A of T, eithe {u, v} A o {u, w} A. We wish to bound the size of a minimal pecolating set A of T which contains {u, w}. If {u, w} A, then A consists of u and a minimal pecolating set of T \ {u, v}, so A 1 + E(T \ {u, v}, 2) 2 + E(T \ {u, v, w}, 2) by Lemma 15. Thus, E(T, 2) = E(T \ {u, v, w}, 2) + 2. Finally, we pove the coectness of ou algoithm fo Case D. Lemma 17. Let T be a tee with a tailing staight pseudo-sta with at most one leaf attached to the cente vetex. Let v be the cente vetex, S the est of the pseudo-sta, and L S the set of leaves. Let T = T \ S u, whee u is a single leaf connected to v. Then given any lagest minimal pecolating set A of T, the method descibed in Case D yields a lagest minimal pecolating set of T. Poof. The point of the poof is that, with espect to pecolation, a tailing pseudo-sta with at most one leaf adjacent to the cente vetex behaves exactly like a tailing P 2. We stat by descibing how a tailing P 2 behaves. Given a minimal pecolating set A on a tee with a tailing P 2, we know that the leaf of the P 2 will be in A. If the P 2 is not intenally spanned, then the leaf will be the only vetex of the tailing P 2 in A. If the P 2 is intenally spanned, then one aditional vetex (namely, the othe vetex of the P 2 ) will be in A. Moeove, if T minus the P 2 is intenally spanned, then the P 2 cannot be intenally spanned. Now we tun ou attention to tailing pseudo-stas. Fist note that thee will be a lagest minimal pecolating set of T in which v is not initially infected. To see this, note fist that v L intenally spans the pseudo-sta, so we need only show that it is possible to choose a lagest minimal pecolating set B of the pseudo-sta with v / B. Howeve, it is clea that we can do this (simply choose any two neighbos of v, including any neighbos which ae leaves). Now, since v has only one neighbo outside of the pseudo-sta and because the pseudosta is staight, any minimal pecolating set A will contain at least one neighbo of v in the pseudo-sta. Since v L infects the entie pseudo-sta and because v is adjacent to at most one leaf, we see that the pseudo-sta will be intenally spanned by A if and only if the electonic jounal of combinatoics 19 (2012), #P64 15

16 pecisely two neighbos of v ae in A. Since v is adjacent to at most one leaf, this means that if A is any minimal pecolating set in which the pseudo-sta is intenally spanned and B is any minimal pecolating set in which the pseudo-sta is not intenally spanned, then A S and B S ae independent of A and B, and A S = B S + 1. This is exactly the case fo tailing P 2 s, and thus, the esult is poven. This concludes the (sketch of) the poof of the coectness of the algoithm ESet. 4 Conclusion Relatively little is known about extemal minimal pecolating sets. We summaize hee what is known and suggest seveal natual poblems to conside. It tuns out that, m(t, ) is usually easie to compute than E(T, ). This pape descibes how to find m fo tees. Exact values of m ae known fo the n n gid in d dimensions [n] d with = 2. In [12] Pete finds estimates fo m(g, ) fo gids. Fo E(G, ), we show hee how to compute it fo tees with = 2, and in [13] E(Q n, 2) is found exactly and shown to be O(2 n/4 ). Additionally Mois [11] showed 4n2 33 E([n]2, 2) n2. 6 This leaves many open questions elated to -bootstap pecolation, and the following suggestions ae by no means exhaustive. We stat by asking a few moe questions about finite tees. Fo a fixed 2, we call a pai (a, b) possible if fo evey ɛ > 0, thee is a tee T of size n such that m(t,) is within ɛ of a and E(T,) is within ɛ of b. n n Question 18. Which pais (a, b) ae possible? Fo example, fo = 2 we can use Poposition 3, Theoem 5, the fact that m(t, 2) E(T, 2), and the fact that E(T, 2) n to see that any possible pai must lie in the convex hull of (1/2, 1/2), (1/2, 3/4), (3/4, 1) and (1, 1). Moeove, using Theoem 4 and the fact that m(t, 2) l we see that E(T, 2) m(t,2) 2 n. Thus, any possible pai must lie 3 3 in the convex hull of (1/2, 1/2), (1/2, 3/4), (5/8, 7/8) and (1, 1). On the othe hand, we know that (1/2, 1/2), (1, 2, 3/4), and (1, 1) ae possible using complete binay tees, the extemal example fo Theoem 5 and stas, and by connecting them with shot paths we know that pais (a, b) in thei convex hull must also be possible. This leaves the convex hull of (1/2, 3/4), (5/8, 7/8), and (1, 1) as an indeteminate egion which could benefit fom futue study. Anothe diection fo futue study would be to look at fine numeical invaiants of tees that the numbe of vetices and numbe of vetices of degee less than, and see what bounds could be poven using these invaiants. Question 19. Is thee a numbe associated to tees (othe than the numbe of vetices of degee less than ) which gives a stonge uppe bound on E(T, )? Does this numbe give us any moe infomation about m(t, )? Anothe natual question to ask is whethe o not it is possible to find an efficient algoithm to compute E(T, ) fo > 2. As we mention, this question will likely be difficult because Lemma 15 no longe holds. the electonic jounal of combinatoics 19 (2012), #P64 16

17 Question 20. Is thee an polynomial-time algoithm fo computing E(T, ) fo > 2? Is thee an O(n) algoithm? Woking in a diffeent diection, it would also be inteesting to ty to apply some of the techniques of this pape to othe types of gaphs with elatively few edges. Question 21. What ae m(g, ) and E(G, ) fo G a unicyclic gaph? Can the esults fo tees be extended to an even moe geneal class of gaphs? Most of the poofs of ou m(g, ) algoithms cay though fo geneal gaphs. Howeve, some of ou E(G, 2) pocedues do not cay though, because the poof of Lemma 15 does not wok fo non-tees. As Balogh, Bollobàs and Mois [5] suggest, we could also conside less spase gaphs which have moe stuctue, such as the hypecube. Question 22. What is m(q n, 3)? What is E(Q n, 3)? Ae thee bounds fo geneal? See the conclusion of [5] fo a summay of what is known, and a conjectue fo geneal. Finally, it would be vey inteesting to continue the wok of Mois [11] and obtain at least asymptotic esults fo E(G, ) fo gids with = 2 and d 2. Question 23. What is E([n] d, ) asymptotically fo = 2 and d? Can anything be said fo geneal? Acknowledgements This eseach was done while I was a student at Note Dame, as a paticipant in the Univesity of Minnesota Duluth math REU, which is un by Joe Gallian. The REU was suppoted by the National Science Foundation and the Depatment of Defense (gant numbe DMS ) and the National Secuity Agency (gant numbe H ). I would like to thank Joe Gallian fo all of his help and encouagement. I would like to thank the pogam advises Nathan Kaplan and Nathan Pfluege fo all thei help, especially fo eading dafts of this pape. I would like to thank all of the Duluth visitos, especially Sasha Ovetsky Fadkin, Phil Matchett Wood and Ricky Liu fo thei help. I would also like to thank Robet Mois at Cambidge fo talking to me about the poblem. Finally, I would like to thank the eviewe fo helping with many aspects of the pape, paticulaly fo suggesting seveal of the open poblems in the conclusion. Refeences [1] J. Adle and U. Lev. Bootstap pecolation: visualizations and applications. Baz. J. Phys., 33: , [2] M. Aizenman and J. L. Lebowitz. Metastability effects in bootstap pecolation. J. Phys. A, 21(19): , the electonic jounal of combinatoics 19 (2012), #P64 17

18 [3] J. Balogh, B. Bollobás, H. Duminil-Copin, and R. Mois. The shap metastability theshold fo -neighbo bootstap pecolation. Tans. Ame. Msth. Soc., to appea. [4] J. Balogh, B. Bollobás, and R. Mois. Bootstap pecolation in thee dimensions. Ann. Pobab., 37: , [5] J. Balogh, B. Bollobás, and R. Mois. Bootstap pecolation in high dimensions. Combin. Pob. Computing, 19: , [6] J. Balogh, Y. Pees, and G. Pete. Bootstap pecolation on infinite tees and nonamenable goups. Combin. Pob. Computing, 15: , [7] R. Cef and E. Ciillo. Finite size scaling in thee-dimensional bootstap pecolation. Ann. Pobab., 27(4): , [8] R. Cef and F. Manzo. The theshold egime of finite volume bootstap pecolation. Stochastic Pocess. Appl., 101(1):69 82, [9] J. Chalupa, P. L. Leath, and G. R. Reich. Bootstap pecolation on a bethe lattice. J. Phys. C., 12:L31 L35, [10] Alexande Holoyd. Shap metastability theshold fo two-dimensional bootstap pecolation. Pobab. Theoy Related Fields, 125(2): , [11] Robet Mois. Minimal pecolating sets in bootstap pecolation. Electon. J. Combin., 16(1):#R2, 20pp, [12] Gabo Pete. Disease pocesses and bootstap pecolation. Thesis fo diploma at the Bolyai Institute, József Attila Univesity, Szeged, [13] Eic Riedl. Lagest minimal pecolating sets in hypecubes unde 2-bootstap pecolation. Electon. J. Combin., 17(1):#R80, 13pp, the electonic jounal of combinatoics 19 (2012), #P64 18