Minimum Power Dominating Sets of Random Cubic Graphs

Transcription

1 Minimum Power Dominating Sets of Random Cubic Graphs Liying Kang Dept. of Mathematics, Shanghai University N. Wormald Dept. of Combinatorics & Optimization, University of Waterloo

2 Outline 1 Introduction 2 Lower Bound for all Cubic Graphs via Bisection Width 3 Random Graphs and Differential Equations 4 Algorithm PD1 and an Upper Bound 58 Algorithm PD2 and an Upper Bound

3 1 Introduction Power Domination Problem Let G be a connected graph and S a subset of its vertices. Denote by M(S) the set monitored by S, defined algorithmically as follows: 1. (domination) M(S) S N(S) 2. (propagation) If there exist any v M(S) and w / M(S) such that N(v) \ M(S) = {w} choose any such w and set M(S) M(S) {w}

4 In other words, the set M(G) is obtained from S as follows. First put into M(S) the vertices from the closed neighbourhood of S. Then repeatedly add to M(S) any vertex w that has a neighbour v in M(S) such that all the neighbours of v, apart from w, are already in M(S). When such a vertex w no longer exists, the construction of M(S) is complete. The set S is called a power dominating set of G if M(S) = V (G), and the power domination number γ P (G) is the minimum cardinality of a power dominating set

5 Haynes et al. (2002) showed that the problem is NP-complete even when the input graph is bipartite; they presented a linear-time algorithm to solve PDS optimally on trees. Kneis et al. (2006) generalized this results to a linear-time algorithm that finds an optimal solution for graphs that have bounded tree-width. Guo, Niedermeier, and Raible (2008) developed a combinatorial algorithm based on dynamic-programming for optimally solving PDS on graphs of tree-width k. The running time of their algorithm is O(c k2 n), where c is a constant

6 Liao and Lee (2005) proved that PDS on split is NP-complete, and they presented a polynomial-time algorithm for solving PDS optimally on interval graphs. Aazami and Stilp (2009) gave an O( n)-approximation algorithm for planar graphs and showed that their methods cannot improve on this approximation guarantee. Dorfling and Henning (2006) computed the power domination number, i.e., the size of an optimal power dominating set, for n m grids. Zhao, Kang and Chang (2006) showed that the size of a minimum power dominating set is at most n/3 for any connected graph G of order n 3, and γ P (G) n/4 for any connected claw-free cubic graph G of order n

7 We consider random cubic graphs that are generated uniformly at random (u.a.r.). We say that a property B = B n of a random graph holds asymptotically almost surely (a.a.s.) if the probability that B holds tends to 1 as n tends to infinity. When discussing any cubic graph on n vertices, we assume n to be even to avoid parity problems. In this paper, we present two heuristics for finding a small power dominating set of cubic graphs. We analyse the performance of these heuristics on random n- vertex cubic graphs using differential equations and obtain two upper bounds on the size of the power dominating set P returned by these algorithms. We show that, for the second heuristic, P asymptotically almost surely satisfies P n. On the other hand, we give the lower bound P 1/29.7n n a.a.s

8 2 Lower Bound for all Cubic Graphs via Bisection Width Define the boundary S to be the set of edges with one end in S and the other end in V (G) \ S. The isoperimetric number of G is defined to i(g) = S min S V (G) /2 S. The bisection width of a graph G with an even number of vertices is bw(g) = min S. S = V (G) /

9 Proposition 1 If G is cubic, then γ P (G) bw(g) 1. 3 Kostochka and Melnikov showed that a random cubic graph G a.a.s. satisfies i(g) n/4.95n + o(n). This immediately implies bw(g) n/9.9 + o(n), which then gives the following. Corollary 2 A random cubic graph G on n vertices a.a.s. satisfies γ P (G) n/ o(n). Note that 1/29.7 > , which is quite close to half of the upper bound featuring as the main result of this paper

10 3 Random Graphs and Differential Equations 3.1. Generating Random Cubic Graphs The model used to generate a cubic graph u.a.r., originally proposed by Bollobas. Take 3n points in n buckets labelled 1,..., n with three points in each bucket Choose u.a.r. a disjoint pairing (i.e. perfect matching) of the 3n points. Obtain a cubic pseudogaph by regarding each bucket as a vertex and each pair as an edge

11 3.2. Analysis Using Differential Equations One method of analyzing the performance of a randomized algorithm is to use a system of differential equations to express the expected changes in variables describing the state of the algorithm during its execution. N. Wormald (1995) gave an exposition of this method; it has been applied to many other combinatorial optimization problems and randomised greedy algorithms. The algorithm will be applied to the random pseudogaph generated by the pairing. During the generation of the pairing for a random cubic graph, we may choose the pairs sequentially. The first point, p i, of a pair may be chosen by any rule, but in order to ensure that the cubic graph is generated u.a.r., the second point, p j, of that pair must be selected u.a.r. from all the remaining free points. The freedom of choice of p i enables us to select it u.a.r. from the vertices of given degree in the evolving graph. Using B(p k ) to denote the bucket that the point p k belongs to, we say that the edge (B(p i ), B(p j )) is exposed when the second point of the pair has been selected

12 In order to analyse our algorithm using a system of differential equation, we incorporate the algorithm into the pairing process that generates the random graph. In this way, we may generate the pairs of the random pairing in precisely the order that the corresponding edges are examined by the algorithm. Throughout the execution of the generation process, vertices will increase in degree until the generation is complete and all vertices have degree 3. We refer to the (part of) the graph which has been generated at any particular time as the evolving graph

13 3.3. Nature of the algorithms The algorithms we use to find a power dominating set P of cubic graphs are greedy algorithms based on selecting vertices of given degree. We say that our algorithms proceeds as a series of operations. We denote the set of vertices of degree i of the evolving graph at time t by V i = V i (t) and let Y i = Y i (t) denote V i. It will turn out that, in order to analyse our algorithms, the only knowledge we require of the state of the evolving graph at any point during the execution is determined by Y 0, Y 1, and Y

14 4 Algorithm PD1 and an Upper bound A Type 1 operation consists of selecting a vertex u u.a.r. from V 1 and then doing the following. Expose the edges incident with u, to vertices v 1, v 2 say). If, as in Figure 1(a) or (b), either of v 1 or v 2 is now in V 2, the operation finishes. So we are left with considering v 1, v 2 V 1. In this case, select v u.a.r. from {v 1, v 2 }, expose all edges incident with v (to vertices w 1, w 2 say) and do the following. If w 1, w 2 V 1, set P P {w 1 }, and expose all edges incident with w 1 ; see Figure 1(c)

15 u v v 2 1 (1a) u u v 1 v 2 (1b) v 1 v 2 w 1 w 2 w z (1e) u u v 1 v 2 v 1 v 2 w 1 w 2 w 1 w 2 w (1c) u v 1 v 2 w 1 w 2 w (1f) Fig. 1. (1d) u v 1 v 2 w 1 w 2 w (1g) z

16 If w 1 V 2, w 2 V 1, expose an edge incident with w 1, to a vertex w (say). If w V 2, do nothing, see Figure 1(d). On the other hand, if w V 1, expose an edge incident with w to a vertex z (say), set P P {z}, and expose all edges incident with z. If w 1 V 1, w 2 V 2, do the same thing with w 1 and w 2 swapped. If w 1, w 2 V 2, expose an edge incident with w 2, to a vertex w (say). If w V 2, do nothing. On the other hand, if w V 1, expose an edge incident with w to a vertex z (say), set P P {z}, and expose all edges incident with z

17 Fig. 2. A Type 2 operation consists of selecting a vertex u u.a.r. from V 2 and exposing an edge incident with u. Since u was in S and had two neighbours in S, it power dominates the new vertex reached. See Figure 2. u

18 Algorithm PD1 select u u.a.r. from V 0 P {u} expose all edges incident with u while (Y 1 + Y 2 > 0) do if (Y 2 > 0) perform the Type 2 operation else perform the Type 1 operation end while

19 4.1. An Upper Bound Theorem 3 A random cubic graph on n vertices asymptotically almost surely has a minimum power dominating set with less than n vertices Preliminary analysis In order to analyse our randomized algorithms for finding a power dominating set P of cubic graphs, we calculate the expected change in this state over one unit of time in relation to the expected change in the size of P. Let P = P (t) denote P at any stage of the algorithm (time t) E X denote the expected change in a random variable X conditional upon the history of the process

20 We then regard E Y i /E P as the derivative dy i /dp, which gives a system of differential equations. The solutions to these equations describe functions which represent the behavior of the variable Y i. There is a general result which guarantees that the solutions of the differential equations almost surely approximate the variables Y i. The expected size of the power dominating set may be deduced from these results

21 Let s = s(t) denote the number of free points available in all buckets at a given time t. Note that s = 2 (3 i)y i. i=0 The expected change in Y i due to changing the degree of an increase from i to i + 1 by exposing one of its incident edges (at time t) is ρ i + o(1), where ρ i = ρ i (t) = (i 3)Y i + (4 i)y i 1, 0 i 2. s

22 Preliminary Equations Operations of Type 2 involve the selection of a vertex v from V 2. The expected changes in Y i for an operation of Type 2 given in Fig. 1 is α i + o(1), where α i = δ i2 + ρ i. (1) The expected increase in P is just E P = 0. (2)

23 We now consider operation of Type 1. The expected change in Y i for operation 1h given in Fig. 2 (at time t) is β hi + o(1), where β ai = 2δ i2 3δ i1, 0 i 2, β bi = δ i2 δ i1 δ i0, 0 i 2, β ci = δ i1 4δ i0 + 2ρ i, 0 i 2, β di = δ i2 δ i1 3δ i0, 0 i 2, β ei = δ i2 4δ i0 + 3Y 0 s (2ρ i δ i0 ) + 2Y 1 s (ρ i δ i1 ), 0 i 2, β fi = 2δ i2 3δ i1 2δ i0, 0 i 2, δ gi = 2δ i2 2δ i2 3δ i0 + 3Y 0 s (2ρ i δ i0 ) + 2Y 1 s (ρ i δ i1 ), 0 i

24 Therefore, the probabilities that operations of type 1a, 1b, 1c, 1d, 1e, 1f, 1g are given by P (1a) = 4Y 1 2 s 2 + o(1), P (1b) = 12Y 0Y 1 s 2 + o(1), P (1c) = 81Y 0 4 s 4 P (1d) = 216Y 0 3 Y1 2 s 5 + o(1), P (1e) = 324Y 0 4 Y1 1 s 5 + o(1), P (1f) = 72Y 0 2 Y1 3 respectively. s 5 + o(1)p (1g) = 108Y 3 s 5 0 Y1 2 + o(1). + o(1),

25 So the expected change in Y i for an Operation Type 1 is β i + o(1), where β i = 4Y 1 2 s 2 β ai + 12Y 0Y 1 s 2 β bi + 81Y 0 4 s 4 β ci + 216Y 0 3 Y1 2 s 5 β di + 324Y 4 s Y 0 2 Y1 3 s 5 β fi + 108Y 0 3 Y1 2 s 5 β gi, 0 i 2. The expected increase in P is just E P = ( 3Y 0 s )2 (1 2 ( 2Y 1 2 )2 3Y 0 s (2Y 1 = 9Y 0 2 s 2 216Y 1 2 Y0 3 s 5 72Y 0 2 Y1 3 s 5. s )3 ) 0 Y 1 β ei

26 We define a birth to be the generation of a vertex in V 2 by processing a vertex of V 1 or V 2. The expected number of births from processing a vertex from V 1 (at time t) is ν 1 + o(1), where ν 1 = 8Y 1 2 s Y 0Y 1 s Y 0 4 Y 1 s Y 0 3 Y1 2 s Y 0 4 Y 1 s 5 6Y 0 + 2Y 1 s + 144Y 0 2 Y1 3 s Y 0 3 Y1 2 s 5 (1 + 6Y 0 + 2Y 1 2Y 1 s s ). 2Y 1 s

27 The expected number of birth from a Type 2 operation (at time t) is ν 2 + o(1), where ν 2 = ν 2 (t) = 2Y 1 s

28 We define a clutch to be a series of operations from an operation of Type 1 up to not in including the next operation of Type 1. Consider the Type 1 operation at the start of the clutch to be the first generation of a birth-death process in which the individual are the vertices in V 2, each give birth to a number of children (essentially independent of the others) with expected number ν 2. Then, the expected number in the jth generation is ν 1 ν j 1 2 and the expected total number of births in the clutch is ν 1 1 ν

29 The equation giving the expected change in Y i for a clutch is given by E Y i = β i + ν 1α i 1 ν 2 + o(1). (3) The equation giving the expected increase in D for a clutch is given by E P = 9Y 0 2 s 2 216Y 1 2 Y0 3 s 5 72Y 0 2 Y1 3 s 5 (4)

30 The Differential Equations Write Y i (t) = nz i (t/n), s(t) = nξ(t/n), ρ i (t) = nψ i (t/n), α i (t) = nτ i (t/n), β i (t) = nϕ i (t/n), ν i (t) = nω i (t/n). Using equation (3), representing E Y i for processing a clutch, suggests the differential equation z i = ϕ i + ω 1 1 ω 2 τ i + o(1). (5)

31 Using equation (4), representing the increasing in the size of P after processing a clutch, and write P (t) = nz(t/n) suggests the differential equation for z as z = 9z2 0 ξ 2 216z3 0z 2 1 ξ 5 72z2 0z 3 1 ξ 5. (6) The solution to these system of differential equations represents the cardinalities of the sets V i and P (scaled by 1/n) for given t. The equations are (5) and (6) with initial conditions z 0 (0) = 1, z 1 (0) = 0, z 2 (0) = 0 and z(0) =

32 We use a result from N. Wormald (1999) to show that, the function representing the solutions to the differential equations almost surely approximate the variable Y i and P with error o(1). Theorem 4 Let Ŵ = Ŵ (n) Ra+1. For 1 l a, where a is fixed, let y l : S (n)+ R and f l : R a+1 R, such that for some constant C 0 and all l, y l (h l ) < c 0 n for all h l S (n)+ for all n. Let Y l (t) denote the random counterpart of y l (h l ). Assume the following three conditions hold, where in (ii) and (iii) W is some bounded connected open set containing the closure of {(0, z 1,..., z a ) : P (Y l (0) = z l n, 1 l a) 0 for some n}. (i) For some functions β = β(n) 1 and γ = γ(n), the probability that max 1 l a Y l (t + 1) Y l (t) β. condition upon H l is at least 1 γ for t < min{t W, TŴ }

33 (ii) For some function λ 1 = λ 1 (n) = 0(1), for all l a, E(Y l (t + 1) Y l (t) H t ) f l (t/n, Y 1 (t)/n,..., Y a (t)/n λ 1 for t < min{t W, TŴ }. (iii) Each function f l is continuous, and satisfies a Lipschitz condition, on W {(t, z 1,..., z a ) : t 0}, with the same Lipschitz constant for each l

34 Then the following are true: (a) For (0, ẑ 1,..., ẑ a ) W the system of differential equations dz l dx = f l(x, z 1,..., z a ), l = 1,..., a has a unique solution in W for z l : R R passing through z l (0) = z l, 1 l a, and which extends to points arbitrarily close to the boundary of W ; (b) Let λ > λ 1 + C 0 nγ with λ = o(1). For a sufficiently large constant C, with probability 1 O(nγ + β λ exp( nλ3 β 3 )), Y l (t) = nz l (t/n) + O(λn) uniformly for 0 t min{σn, TŴ } and for each l, where z l (x) is the solution in (a) with ẑ l = 1 n Y l(0), and σ = σ(n) is the supremum of those x to which the solution can be extended before reaching within l -distance Cλ of the boundary of W

35 By Theorem 4, the random variables Y i /n and P/n a.a.s. remain within o(1) of the corresponding deterministic solution to the differential Eqs. (5) and (6) until a point arbitrarily close to where it leaves the set W, or until t = TŴ if that occurs earlier. We compute the ratio dz i /dz, and we have dz i dz = ϕ i + ω1 1 ω 2 τ i, i {0, 1, 2} 9z0 2 ξ 216z3 2 0 z2 1 ξ 72z2 5 0 z3 1 ξ 5 where, differentiation is with respect to z and all function can be taken as functions of z. By solving this we find that the solution hits a boundary of W at ξ = ε. The differential equations were solved using a Runge-Kutta method, giving ξ = ε at z < This corresponds to the size of the power dominating set (scaled by 1 n ) when all vertices are used up, thus proving the theorem

36 5 Algorithm PD2 and an Upper Bound In this section we present Algorithm PD2 in order to improve the bound obtained using Algorithm PD1.In Algorithm PD2, H denote the set of vertices of degree 1 of the evolving graph at time t and will be dominated from below. F denote the set of vertices of degree 2 of the evolving graph at time t and next below is in V 2 or will be dominated from below. A Type 1 operation consists of selecting a vertex u u.a.r. from V 1 and then doing the following. Expose the edges incident with u, to vertices v 1, v 2 (say). If, as in Fig 3(a) or (b), either of v 1 or v 2 is now in V 2, the operation finishes. If v 1, v 2 V 1, label v 1 as H, H H {v 1 }. See Figure 3(c)

37 v 1 u v 2 3(a) v 1 u v 2 3(b) v 1 H u v 2 3(c) Fig

38 u Fig. 4 A Type 2 operation consists of selecting a vertex u u.a.r. from V 2 and expose an edge incident with u. See Figure 4. A Type 3 operation consists of selecting a vertex u u.a.r. from F and exposing an edge incident with u, to vertices v (say). If v V 2, the operation finishes. If v V 1, label v as H, H H {v}. See Figure

39 v u F u v H F 5(a) Figure 5 5(b)

40 A Type 4 operation consists of selecting a vertex u u.a.r. from H and then doing the following. Expose the edges with u, to vertices v 1, v 2 (say). If, as in Fig 6(a) or (b), v 1 V 2 v 2 V 2, label v 2 as F, F F {v 2 }. If v 1 V 2 v 2 V 1, label v 1 as F, F F {v 1 }. So we are left with considering v 1, v 2 V 1. In this case, select v u.a.r. from {v 1, v 2 }, expose all edges incident with v to vertices w 1, w 2 (say) and do the following. If w 1 V 2 w 2 V 2, label w 1 as F, F F {w 1 }. See Figure 6(c) If w 1 V 2 w 2 V 1, P P {w 2 }, expose edges incident with w 2. See Figure 6(d). If w 1 V 1 w 2 V 1, label w 1 as H, H H {w 1 } P P {w 2 }, expose edges incident with w 2. See Figure 6(e)

41 H u v v 2 1 6(a) F H u v 1 v 2 H 6(b) u v 1 v 2 w 1 w 2 F w 1 F u v 1 v 2 w 2 w 1 H H H u v 1 v 2 w 2 6(c) (d) Figure 6 6(e)

42 5.1. Algorithm PD2 select u u.a.r. from V 0 P {u} expose all edges incident with u while (Y 1 + F + H > 0) do if ( F > 0) perform the Type 3 operation. else if H > 0, perform the Type 4 operation. else if Y 1 > 0, perform the Type 1 operation. else perform the Type 2 operation. end while

43 5.2. The upper Bound We analyse the combined algorithm and pairing process using differential equations and in this way prove the following theorem. Theorem 5 A random cubic graph on n vertices asymptotically almost surely has a minimum power dominating set with less than n vertices

44 Preliminary Equations Operations of Type 1 involve the selection of a v from V 1. The expected changes in Y i for an operation of Type 1 given in Fig.1 is β i + o(1), where β i = 4Y 1 2 s 2 (2δ i2 3δ i1 ) + 12Y 0Y 1 s 2 (δ i2 δ i1 δ i0 ) + 9Y 0 2 s 2 (δ i1 2δ i0 ). The expected increase in P is just E P =

45 Operations of Type 2 involve the selection of a v from V 2. The expected changes in Y i for an operation of Type 2 given in Fig.1 is α i + o(1), where α i = δ i2 + ρ i. The expected increase in P is just E P =

46 The expected change in Y i for an operation of Type 3 is γ i + o(1), where The expected increase in P γ i = 2Y 1 s ( δ i1) + 3Y 0 s ( δ i2 + δ i1 δ i0 ). E P =

47 The expected change in Y i for an operation of Type 4 is θ i + o(1), where θ i = 4Y 1 2 s 2 (2δ i2 3δ i1 ) + 12Y 1Y 0 s 2 (δ i2 δ i1 δ i0 ) + 36Y 1 2 Y0 2 s 4 (2δ i2 2δ i1 2δ i0 ) + 108Y 1Y0 3 s 4 (δ i2 δ i1 3δ i0 + 2ρ i ) + 81Y 0 4 s 4 (δ i1 4δ i0 + 2ρ i ). The expected increased in P is E P = 108Y 1Y0 3 s Y 0 4 s

48 The expected number of vertices F, H, V 2 from processing a vertex from V 1 is ν 1f + o(1), ν 1h + o(1), ν 12 + o(1) respectively, where ν 1f = 0, ν 1h = ( 3Y 0 s ν 12 = ( 2 2Y 1 s ) 2 = 9Y 2 0 s 2, ) 2 3Y s (2Y 1 s ) = 8Y 2 1 s Y 1Y 0 s 2. The expected number of vertices F, H, V 2 from processing a vertex from V 2 is ν 2f + o(1), ν 2h + o(1), ν 22 + o(1), where ν 2f = ν 2h = 0, ν 22 = 2Y 1 s

49 The expected number of vertices F, H, V 2 from processing a vertex from F is ν ff + o(1), ν fh + o(1), ν f2 + o(1) respectively, where ν ff = 0, ν fh = 3Y 0 s, ν f2 = 2Y 1 s. The expected number of vertices F, H, V 2 from processing a vertex from H is ν hf + o(1), ν hh + o(1), ν h2 + o(1) respectively, where ν hf = ( 2Y 1 s ν hh = ( 3Y 0 s ν h2 = ( 2Y 1 s ) 2 (2Y s ) 4 81Y 4 = 0 s 4, ) 2 + (3Y 0 s ) (3Y 0 s ) 2 (2Y 1 s ) + (2Y 1 s ) (3Y 0 s ) 2 (3Y 0 ) 3 4Y = 1 2 s s Y 1Y 0 ) 3 2Y 1 s ( Y 1 s s Y 1 2 Y0 2 s 4, ) + (3Y 0 s ) 4 (4Y 1 ). s

50 We define a clutch to be a series of operations from an operation of Type 1 up to not in including the next operation of Type 1. Let the expected total number of vertices V 2, F, H in the clutch is a, b, c, respectively. Let matrix ( ) D = and ν ff ν fh ν hf ν hh, Then det = (1 ν hh ) (1 ν ff ) ν hf ν fh. ( (I D) 1 = 1 det 1 ν hh ν fh ν hf 1 ν ff ),

51 So we have (b, c) = (ν 1f, ν 1h ) + (ν 1f, ν 1h ) D + (ν 1f, ν 1h ) D = (ν 1f, ν 1h )(I + D + D ) = (ν 1f, ν 1h )(I D) 1 = ( 0, 9Y 0 2 ) (I D) 1 s 2 = ( 9Y0 2 s 2 det ν hf, 9Y0 2 ). s 2 det b = 9Y 0 2 s 2 det ν hf, c = 9Y 0 2 s 2 det

52 a = (ν 12 + ν 12 ν 22 + ν 12 ν ) + b (ν f2 + ν f2 ν 22 + ν f2 ν ) +c (ν h2 + ν h2 ν 22 + ν h2 ν ) = (ν 12 + b ν f2 + c ν h2 )(1 + ν 22 + ν ) = ν 12 + b ν f2 + c ν h2 1 ν 22. The equation giving the expected change in Y i for a clutch is given by E Y i = β i + a α i + b γ i + c θ i. (7) The equation giving the expected increasing in P for a clutch is given by E P = c (108Y0 3 Y 1 s Y 0 4 ). (8) s

53 The Differential Equations Using equation (7), representing E Y i for processing a clutch,suggests the differential equation z i = ϕ i + A τ i + B ζ i + C η i. (9) Using equation (8), representing the increasing in the size of P after processing a clutch, and write P (t) = nz(t/n) suggests the differential equation for z as z = C ( 108z 3 0 z 1 ξ z4 0 ξ 4 ). (10)

54 The solution to these system of differential equations represents the cardinalities of the sets V i and P (scaled by 1/n) for given t. The equations are (7) and (8) with initial conditions z 0 (0) = 1, z 1 (0) = 0, z 2 (0) = 0 and z(0) = 0. Using the similar discussion as in Section 3, we compute the ratio dz i /dz, and we have dz i dz = ϕ i + A τ i + B ζ i + C η i C (108z 0 3z ), i {0, 1, 2} 1 ξ + 81z4 4 0 ξ 4 where, differentiation is with respect to z and all function can be taken as functions of z. By solving this we find that the solution hits a boundary of W at ξ = ε. The differential equations were solved using a Runge-Kutta method, giving ξ = ε at z < This corresponds to the size of the power dominating set (scaled by 1 n ) when all vertices are used up, thus proving the theorem

55 谢谢大家!