Surviving rates of trees and outerplanar graphs for the

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1 Survivig rates o trees ad outerplaar graphs or the ireighter problem Leizhe Cai Yogxi Cheg Elad Verbi Yua Zhou Abstract The ireighter problem is a discrete-time game o graphs itroduced by Hartell i a attempt to model the spread o ire, diseases, computer viruses ad suchlike i a macro-cotrol level. To measure the deece ability o a graph as a whole, Cai ad Wag deied the survivig rate o a graph G or the ireighter problem to be the average percetage o vertices that ca be saved whe a ire starts radomly at oe vertex o G. this paper, we prove that the survivig rate o every -vertex outerplaar graph is at least Θ( log ), which is asymptotically tight. We also show that the greedy strategy o Hartell ad Li or trees saves at least Θ( log ) percetage o vertices o average or a -vertex tree. Key words: ireighter problem, survivig rate, tree, outerplaar graph troductio The study o a discrete-time game, the ireighter problem, was iitiated by Hartell [7] i 995 i a attempt to model the spread o ire, diseases, computer viruses ad suchlike i a macro-cotrol level. At time 0, a ire breaks out at a vertex v o a graph G = (V, E). At each subsequet time, a ireighter protects oe vertex ot yet o ire, ad the ire the spreads rom all burig vertices to all their uprotected eighbors. The process eds whe the ire ca o loger spread. Oce a vertex is burig or protected, it remais so durig the whole process, ad i the ed all vertices that are ot burig are saved. A geeral objective o the ireighter is to save as may vertices as possible. Fibow et al. [4] showed that it is NP-hard or the ireighter to save the maximum umber o vertices, eve or trees o maximum degree three, but polyomial-time solvable Departmet o Computer Sciece ad Egieerig, The Chiese Uiversity o Hog Kog, Shati, Hog Kog SAR, Chia. address: lcai@cse.cuhk.edu.hk. Departmet o Computig Sciece, Uiversity o Alberta, Edmoto, Alberta T6G E8, Caada. address: chegyx@gmail.com. The stitute or Theoretical Computer Sciece, Tsighua Uiversity, Beijig 00084, Chia. address: {elad.verbi, timzhouyua}@gmail.com.

2 or graphs o maximum degree three with the ire startig at a vertex o degree at most two. MacGillivray ad Wag [9] gave a 0- iteger programmig ormulatio o the problem or trees ad solved the problem i polyomial time or some subclasses o trees, ad recetly Cai et al. [] cosidered several algorithmic issues cocerig ireightig o trees. Develi ad Hartke [3], Fogarty [6], ad Wag ad Moeller [] cosidered a variatio that allows more tha oe ireighters, ad studied the umber o ireighters required to cotai the ire or d-dimesioal grids. Recetly, Ng ad Ra [0] ivestigated a geeralizatio o the ireighter problem o two dimesioal iiite grids whe the umber o ireighters available per time step is ot a costat but periodic. Other variatios o the problem have also bee cosidered i the literature; or istace, to put out the ire as quickly as possible, or to save a give subset o vertices. We reer the reader to a recet survey by Fibow ad MacGillivray [5]. For a vertex v i G, let s(v) deote the maximum umber o vertices i G the ireighter ca save whe the ire breaks out at v. To mesure the deece ability o a -vertex graph G as a whole, Cai ad Wag [] deied the survivig rate ρ(g) o G to be the average percetage o vertices that ca be saved whe the ire starts radomly at oe vertex o v V s(v) the graph, i.e., ρ(g) =. They showed that ρ(t ) or every tree T, ρ(g) > /6 or every outerplaar graph G, ad ρ(h) > 0.3 or every Hali graph H with at least 5 vertices. They also cojectured that ρ(t ) Θ( log ) or every tree T ad asked whether ρ(g) teds to or ay -vertex outerplaar graph whe teds to iiity. this paper, we study survivig rates o trees ad outerplaar graphs. We show i Sectio that the greedy strategy o Hartell ad Li [8] or trees o average saves Θ( log percetage o vertices, ad i Sectio 3 we obtai the asymptotically tight boud Θ( log or the survivig rate o outerplaar graphs. Our results coirm a cojecture o Cai ad Wag [] ad also aswer two o their questios i airmative. the paper, log is o base, ad or a graph G we use G to deote the umber o vertices o G. A outerplaar graph is a graph that ca be embedded o the plae without crossig edges ad with all vertices o the boudary o the exterior ace. Fireightig o trees For the ireighter problem o trees, the ollowig greedy method o Hartell ad Li [8] achieves approximatio ratio / or the umber o saved vertices: the ireighter always protects a vertex that cuts o the maximum umber o o-burig vertices rom the ire. this sectio, we prove that their greedy method o average saves Θ( log ) percetage o vertices, thus achieves a approximatio ratio o Θ( log ) or the survivig rates o trees. This also settles the cojecture o Cai ad Wag [] that the survivig rate o a tree is at least Θ( log ). O the other had, we also costruct a tree whose survivig rate is at most Θ( log ). Let T be a tree. The greedy method o Hartell ad Li produces a strategy or the ireighter, which will be called a HL-strategy or T. Note that a HL-strategy or T is ot ) )

3 T T ( v ) r v r v vi v i v v u vi vi (a) (b) Figure. (a): All ire-sources o v are o the (r, v)-path; (b): T (v i ) T (v i+ ). uique, sice accordig to the greedy method at oe time step there may be more tha oe vertex that the ireighter ca choose to protect. A vertex u is a ire-source or a vertex v i whe the ire starts at u, the greedy method o Hartell ad Li caot always save v, i.e., there is a HL-strategy that will ot save v. Theorem. For a tree T, the greedy method o Hartell ad Li saves o average at least Θ( log ) percet o vertices whe the ire starts radomly at oe vertex o T. Proo: To prove the theorem, we eed oly prove that o vertex v T has more tha 3 + log ire-sources. Let r be a ire-source or v that is arthest away rom v i T, ad P = v 0, v,..., v k the (r, v)-path i T, where v 0 = r ad v k = v. Regard T as a rooted tree with root r, ad deote the subtree rooted at vertex x by T (x). We irst show that all ire-sources or v are o the (r, v)-path P. Suppose that there is a ire-source u P or v. The u is i T (v ) as ay vertex ot i T (v ) is arther away rom v tha r. Sice r is a ire-source or v, some HL-strategy will ot protect v whe a ire starts at r. Thereore T (v ) < / ad hece T T (v ) > /, which implies that T T (u) > / as T T (u) cotais T T (v ). This idicates that, whe a ire starts at u, ay HL-strategy would have saved the paret o u ad hece v, a cotradictio to u beig a ire-source or v (see Figure (a)). Next we show that i v i is a ire-source or v the T (v i ) > T (v i+ ), where i k. Cosider the situatio whe the ire starts at vertex v i. Sice T T (v i ) > / (ote that T T (v i ) cotais T T (v )), ay HL-strategy will protect v i at time. By the assumptio that v i is a ire-source or v, we see that there is a HL-strategy that does ot protect v i+ but aother gradchild v i o v i at time. This implies that T (v i ) T (v i+ ) ad hece T (v i ) > T (v i+ ) (see Figure (b)). Now let v s(0), v s(),..., v s(t) P be ire-sources o v ordered rom r to v. The T (v s(i) ) > 3

4 T (v s(i+) ) as T (v s(i+) ) is a subtree o T (v s(i)+ ). Thereore T (v s() ) > T (v s(3) ) > > i T (v s(i+) ) >, which implies t + log as T (v s() ) < ad hece the lemma. The above theorem aswers a questio o Cai ad Wag [] i airmative that the greedy method o Hartell ad Li achieves approximatio rate Θ( log ) or the survivig rate o a tree, ad also settles the ollowig cojecture o Cai ad Wag regardig survivig rates o trees. Corollary. The survivig rate o every tree is at least Θ( log ). act, the above lower boud or survivig rates o trees is asymptotically best possible, which is established by the ollowig theorem. Theorem.3 Let T h be a balaced complete terary tree (i.e., each o-lea vertex has three childre) o height h ad with vertices. The ρ(t h ) Θ( log ). Proo: We will prove the ollowig: the ire starts at a vertex v o height k (0 k h), let T k deote the subtree with v as its root, the the umber o burt leaves o T k i the ed is at least (3k + ) uder ay protectig strategy. Sice, the umber o vertices o T h, is h i=0 3i = 3h+ = Θ(3 h ), this implies that o matter what protectig strategy is adopted, whe the ire starts radomly at oe vertex o T h, the miimum average percetage o vertices that will get burt i the ed is at least h k=0 (3k + ) 3 h k h k=0 3 h = (h + )3h = Θ( h ) = Θ(log ). which implies the theorem. what ollows we cosider the subtree T k with root v, ad assume that the ire starts at v. The, withi T k the ire stops to propagate at time k. Thus, we ca assume without loss o geerality that the umber o protected vertices i T k is at most k. Furthermore, at time i (0 i k), the ire stops to propagate amog the vertices i T k havig distace at most i rom the root v, thereore we ca assume that there are at most i protected vertices i T k which are withi distace i rom v. Let a j deote the umber o protected vertices i T k that have distace j rom v, j = 0,,, k. The, i j=0 a j i, or i = 0,,, k. the ire starts at v, or each lea u o T k which is saved i the ed, there must exist a acestor w o u such that w is i T k ad w is protected at some time t, where t k. Thereore, the total umber o leaves o T k that are saved i the ed caot exceed k j=0 a j 3 k j. Deie b i = i j=0 a j, the b i i or i = 0,,, k ad thus k a j 3 k j = j=0 k k (b j b j ) 3 k j = b k + b j (3 k j 3 k j ) j= j= j= k k + j (3 k j 3 k j ) = 3k. 4

5 Thereore, the total umber o burt leaves o T k is at least 3 k 3k completes the proo o the theorem. 3 Fireightig o outerplaar graphs = 3k +, which this sectio we prove that the survivig rate o outerplaar graphs is also Θ( log ), which is asymptotically tight as the same rate is asymptotically tight or trees (Theorem.3), ad it is ot hard to veriy that outerplaar graphs orm a superset o trees. For this purpose, we eed oly cosider maximal outerplaar graphs, i.e., outerplaar graphs where the additio o ay edge will destroy the outerplaarity. Let G = (V, E) be a maximal outerplaar graph, i.e., a plaar embeddig o a maximal outerplaar graph with all vertices o the boudary o the exterior ace. We will establish our result or G by cosiderig the dual graph G = (V, E ) o G costructed as ollows: place a vertex iside each ace o G, ad, i two aces have a edge e i commo, joi their correspodig vertices by a edge e crossig oly e. The ireightig problem o vertices o G ca be trasormed ito that o aces o the dual graph G : A ire starts at a ace o G, ad spreads rom a burig ace to each uprotected ace sharig a commo edge with i oe uit o time. each uit o time, a ireighter ca protect oe ace ot yet o ire. See Figure or a example. Let x deote the vertex i G correspodig to the exterior ace o G. t is well kow that G x is a tree o maximum degree 3 as every ace o G (except the exterior ace) is a triagle. We tur G x ito a rooted biary tree T = (V T, E T ) by pickig up a lea r as the root (see Figure (b) or a example). Note that each edge i G T coects a vertex v o T with vertex x, ad or coveiece, we also regard vertex x as a child o v. For each vertex v V T \ {r}, all vertices o the (r, v)-path are acestors o v. For each vertex v V T \ {r}, we use p(v) to deote its paret i T, T v the subtree rooted at v, ad depth(v) the depth o v which is the distace rom the root to v. We also desigate let ad right childre o each vertex v V T \ {r} i a atural way: we start with edge vp(v) ad tur aroud v clockwise i a very small circle, the irst edge we cross coects v with its right child, ad the other child o v is its let child. We will use T to desig our strategy or the dual graph G, ad we will deie a ew more terms beore we ca describe our strategy. A vertex v V T \ {r} is a heavy vertex i T v cotais more tha hal vertices o T p(v), ad a light vertex otherwise. Clearly, each vertex has at most oe heavy vertex as its child, ad every path rom the root has at most log light vertices. A path P (v 0 ) = v 0, v,..., v t is a heavy path i each v i, i t, is a heavy vertex (vertex v 0 ca be either a heavy or light vertex), ad a vertex v i, i t, is a turig vertex i v i is the let (right respectively) child o v i ad v i+ is the right (let) child o v i. Whe desigatig the let ad right child o a vertex v V T \ {r}, we also take the exterior vertex x ito accout. That is, i v has oe (zero respectively) child i T, the there must be oe (two) edge(s) i G coectig v ad x, we also regard x as a child o v. For example i Figure (b), e is the right child o s, d is the let child o c, ad x is both the let ad right child o. 5

6 h l 0 5 k 4 m e g 3 c j (a) The maximal outerplaar graph G (dashed edges) d i b 3 o 4 a q 0 s p 5 9 r a 4 3 b s e c 9 g d 0 8 r q p i h 8 x 7 6 k l m 5 (b) The dual G. G x orms a rooted biary tree with root r. Figure. A maximal outerplaar graph G ad its dual G. For example, a ire startig at vertex 7 i G is equivalet to a ire startig at ace 7 i G. Ater oe time uit the ire will spread to the ace-eighbors 6, 5, 3, 0, 6, 9 ad 8 o ace 7 i G, ad the ireighter ca protect oe o these eighbors (or ay ace other tha 7). For a subtree T v o T, we use Tv x to deote the iduced subgraph G [V (T v ) {x}]. For each ace i G, its top vertex, deoted by v, is the vertex o the boudary o with miimum depth i T. Note that, except root r, each vertex i T is a top vertex o exactly oe ace. Fially we deie two importat aces or : or the two aces eclosig Tp(v x ) i G, the oe that cotais both v ad p(v ) i its boudary is the critical ace o ad the other oe is the early-critical ace o (see Figure 3(a)). Note that i we protect the critical ad ear-critical aces o i sequece, the whole subgraph Tp(v x ) will be cut o rom the rest o T. We are ow ready to describe our strategy or the ireightig i the dual graph G. Let be the ace that starts the ire. Our strategy cosists o at most our rouds o protectio: the irst two rouds will cotai the ire i Tp(v x ), ad the ext two rouds will go alog the heavy path P (p(v )) to save all aces iside a heavy subgraph o Tp(v x ). The strategy is give by the ollowig 4 rules carried out i order. Rule We do othig i depth(v ), otherwise we use Rules -4. Rule For the irst two rouds, we protect the critical ad early-critical aces o i tur 6 4 j 3 9 o 0

7 p(v ) v q w p(v ) v 3 p(v ) v w 4 w 5 (a) the irst two rouds, we use Rule to protect critical ad early-critical aces ad o. P(p(v )) (b) the 3rd roud, we use Rule 3 to protect ace 3. P(p(v )) (c) the 3rd ad 4th rouds, we use Rule 4 to protect aces 4 ad 5. Figure 3. Our strategy whe the ire starts at. Grey aces are protected. The Roma umber i a ace idicates the earliest time or the ace to catch ire. to cotai the ire iside subgraph T x p(v ) (see Figure 3(a)). Rule 3 the heavy path P (p(v )) has at least 6 vertices ad o turig vertex amog the irst 5 vertices, let w be the its 6th vertex. the 3rd roud, we protect the ace whose top vertex is p(w) to save all aces iside subgraph T x w (see Figure 3(b)). We do othig i the 4rd roud. Rule 4 the coditio i Rule 3 does t hold (thus the strategy i Rule 3 has t bee carried out), ad i the heavy path P (p(v )) has at least 9 turig vertices, the let w ad w be the 8th ad 9th turig vertices. Let 4 ad 5 be the aces whose top vertices are p(w ) ad p(w ). We protect 4 ad 5 i the 3rd ad 4th rouds to save all aces iside subgraph Tw x (see Figure 3(c)). t is easy to see that the strategy is valid as it does ot protect ay burig aces. A ace is a ire-source or g i whe the ire starts at ace, our strategy may ot save ace g. A vertex u is a bad-acestor o a ace g i there is a ire-source or g with depth(v ) > such that v is a child o u. We ow show that our strategy esures that the average umber o burt aces i G is O(log ). For this purpose, we irst put a upper boud o the umber o bad-acestors. Lemma 3. Every ace g has at most O(log ) bad-acestors. Proo: By Rule, we kow that whe the ire starts at a ace with depth(v ) >, all aces outside Tp(v x ) are saved because o the protectio o the critical ad early-critical aces o i the irst two rouds. Thereore ay bad-acestor o g is a acestor o v g, ad hece is o the (r, v g )-path P o T. 7

8 r sectio mii-sectio v g Figure 4. Decompositio o the (r, v g )-path ito sectios, ad urther ito mii-sectios. To put a upper boud o the umber o bad acestors o g i P, we irst decompose P ito sectios. Because P cotais at most log light vertices, the deletio o these light vertices divides P ito at most log + sectios. We add each light vertex to the sectio immediately ollows it. Let S be a arbitrary such sectio. The turig vertices o S urther divide S ito sectios, which we reer to as mii-sectios (see Figure 4). We add each turig vertex to the ed o the mii-sectio immediately ollows it. Suppose v P is a bad-acestor o g. By Rule 4, we see that v ca oly reside i the last 9 mii-sectios or each sectio. For each mii-sectio, we see rom Rule 3 that v ca oly be oe o the last 4 vertices i the mii-sectio. t ollows that the umber o possible bad-acestors o g i each sectio is at most 36, implyig that the total umber o possible bad-acestors o P is at most 36( log + ), which completes the lemma. With Lemma 3. at had, we ca ow establish the survivig rate o outerplaar graphs. Theorem 3. The survivig rate o every -vertex outerplaar graph is Θ( log ). Proo: As metioed earlier, the ireightig problem o vertices o a outerplaar graph G is equivalet to that o aces o its dual graph G. From Rules ad, we see that every ire-source or a ace g satisies either (a) depth(v ) or (b) p(v ) is a acestor o v g. There are at most 4 aces satisyig (a), ad the umber o ire-sources satisyig (b) is at most twice the umber o bad-acestors o g as each vertex i T has at most two childre. Sice the umber o bad-acestors o g is at most O(log ) (by Lemma 3.), the umber o ire-sources or g is o more tha 4 + O(log ) = O(log ). Thereore our strategy saves at least O(log ) vertices o average, ad the theorem easily ollows rom this ad Theorem.3. 8

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