Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees

0.13 MB·

by  Michael Gentner

Dynamic Monopolies for Degree Proportional Thresholds
in Connected Graphs of Girth at least Five and Trees
Michael Gentner and Dieter Rautenbach
Institute of Optimization and Operations Research
Ulm University, Ulm, Germany
michael.gentner,dieter.rautenbach @uni-ulm.de
6
{ }
1
0
2
Abstract
b
e
F Let G be a graph, and let ρ [0,1]. For a set D of vertices of G, let the set Hρ(D)
∈
arise by starting with the set D, and iteratively adding further vertices u to the current
5
1 set if they have at least ρdG(u) neighbors in it. If Hρ(D) contains all vertices of G, then
⌈ ⌉
D is known as an irreversible dynamic monopoly or a perfect target set associated with

]
O the threshold function u ρdG(u) . Let hρ(G) be the minimum cardinality of such an
7→ ⌈ ⌉
C irreversible dynamic monopoly.
. ForaconnectedgraphGofmaximumdegreeatleast 1,Chang(Triggeringcascadeson
h ρ
undirected connected graphs, Information Processing Letters 111 (2011) 973-978) showed
t
a
h (G) 5.83ρn(G), which was improved by Chang and Lyuu (Triggering cascades on
m ρ
≤
strongly connected directed graphs, Theoretical Computer Science 593 (2015) 62-69) to
[
h (G) 4.92ρn(G). We show that for every ǫ > 0, there is some ρ(ǫ) > 0 such that
ρ
2 ≤
h (G) (2 + ǫ)ρn(G) for every ρ in (0,ρ(ǫ)), and every connected graph G that has
v ρ ≤
9 maximumdegreeatleast 1ρ andgirthatleast5. Furthermore,weshowthathρ(T)≤ ρn(T)
9 for every ρ in (0,1], and every tree T that has order at least 1.
0 ρ
2
0 Keywords: Irreversible dynamic monopoly; perfect target set
.
1 MSC2010: 05C69
0
6
1 1 Introduction
:
v
i
X We consider finite, simple, and undirected graphs, and use standard terminology and notation.
r Let G be a graph with vertex set V(G). Let φ : V(G) N be a threshold function such
a → 0
that φ(u) is at most the degree d (u) of u in G for every vertex u of G. For a set D of vertices
G
of G, let H (D) be the smallest set D¯ of vertices of G such that D D¯, and every vertex u
(G,φ)
in V(G) D¯ has less than φ(u) neighbors in D¯. Note that the set H ⊆(D) can be constructed
(G,φ)
\
by starting with the set D, and iteratively adding further vertices u to the current set if they
have at least φ(u) neighbors in it. Such iterative expansion processes have been considered in a
variety of contexts [2,6–9,13,15,18,21]. If H (D) = V(G), then D is a φ-dynamic monopoly
(G,φ)
of G. Let h (G) be the minimum cardinality of a φ-dynamic monopoly of G.
φ
By a simple probabilistic argument, very similar to the one used by Alon and Spencer [4] to
prove the Caro-Wei bound on the independence number of a graph [6,20], Ackerman, Ben-Zwi,
Wolfovitz [1] showed
φ(u)
h (G) . (1)
φ ≤ d (u)+1
G
u∈V(G)
X
1
Essentially the same bound was obtained independently by Reichman [19]. For an application
of this argument to independence in hypergraphs see [5].
It is easy to see [9] that (minimum) dynamic monopolies and (maximum) generalized degen-
erate induced subgraphs are dual notions, that is, (1) can be considered the dual counterpart
of bounds as in [3].
A very natural choice for the threshold function is to assign values that are proportional to
the vertex degrees. Specifically, for some real parameter ρ in [0,1], let
φ : V(G) N : u ρd (u) .
ρ 0 G
→ 7→ ⌈ ⌉
If D is a random set of vertices of G that contains each vertex independently at random with
probability ρ, then, for every vertex u of G, the expected number of neighbors of u that belong
to D is ρd (u). This suggests that h (G) might be only slightly bigger than ρn(G), where
G φρ
n(G) is the order of G. Without any restriction on ρ or G though, this intuition is misleading.
In fact, if ρ is positive but much smaller than 1 , then h (G) is at least 1, while ρn(G) can
n(G) φρ
be arbitrarily small. As observed by Chang [10], it is reasonable to consider only values of ρ
that are at least 1 , where ∆(G) is the maximum degree of G, because φ = φ for every
∆(G) 1 ρ
∆(G)
ρ in 0, 1 . For a connected graph G and ρ 1 ,1 , Chang [10] proved
∆(G) ∈ ∆(G)
(cid:16) i h i
h (G) (2√2+3)ρn(G) 5.83ρn(G), (2)
φρ ≤ ≈
which was improved by Chang and Lyuu [11] to
h (G) 4.92ρn(G). (3)
φρ ≤
Note that the bound in (1) might evaluate to Ω(n(G)), because, for instance, vertices of degree
1 contribute 1 rather than O(ρ) to the right hand side of (1). In fact, especially for small values
2
of ρ, and graphs with many vertices of small degrees, the bound (3) can be much better than
the bound (1).
The proof strategies for (2) and (3) are quite different. The bound (2) is proved by a
suitable adaptation of the argument of Ackerman, Ben-Zwi, Wolfovitz [1]. Vertices of small
degree, that is, at most 1, are treated differently from those of large degree, that is, more than
ρ
1. A small set X of vertices of large degree ensures that the remaining vertices of large degree
ρ 0
have few neighbors of small degree outside of H (X ). This allows to apply the argument
(G,φρ) 0
of Ackerman et al. to the vertices of large degree outside of X . The bound (3) is proved
0
by a random procedure that considers a sequence X ,X ,... of random sets of vertices each
1 2
containing every individual vertex independently at random with probability 3.51ρ. Starting
with the empty set, a φ -dynamic monopoly is constructed by iteratively adding the vertices
ρ
in X H (X ... X ) to the current set. Chernoff’s inequality is used to ensure that
i \ (G,φρ) 1∪ ∪ i−1
H (X ... X ) grows sufficiently fast. The proof of (3) has some resemblance to iterative
(G,φρ) 1∪ ∪ i
random procedures that are used to show lower bounds on the independence number [14,16].
It is a natural to ask for the best-possible constant in bounds of the form (2) and (3). We
contribute to this question by showing the following results.
Theorem 1 For every positive real ǫ, there is some positive real ρ(ǫ) such that
h (G) (2+ǫ)ρn(G) (4)
φρ ≤
for every ρ in (0,ρ(ǫ)), and every connected graph G that has maximum degree at least 1 and
ρ
girth at least 5.
2
The proof of Theorem 1 is based on a combination of the techniques from [10,11]. Note that
(5) requires a sufficiently small value of ρ, but that bounds like (2), (3), and (5) are especially
interesting for small values of ρ. It is possible to generalize (5) to strongly connected directed
graphs similarly as in [11].
Theorem 2 If ρ is in (0,1], and T is a tree of order at least 1, then
ρ
h (T) ρn(T). (5)
φρ ≤
Note that h (T) can be computed in linear time [7] for a given tree.
φρ
For many more references to and discussion of related work see [10,11].
2 Proofs of Theorem 1 and Theorem 2
We recall two tools from probability theory.
Markov’s inequality (cf. Theorem 3.2 in [17])
•
If X is a non-negative random variable, and t > 0, then P X t 1E X .
≥ ≤ t
Chernoff’s inequality (cf. Theorem 4.2 in [17]) (cid:2) (cid:3) (cid:2) (cid:3)
•
If Z ,...,Z are independent random variables, and p ,...,p in (0,1) are such that
1 n 1 n
P[Z = 1] = p and P[Z = 0] = 1 p for i [n], then
i i i i
− ∈
P Z1 + +Zn < (1 δ)E Z1 + +Zn < e−δ22E Z1+···+Zn
··· − ···
h i (cid:2) (cid:3)
(cid:2) (cid:3)
for every 0 < δ 1.
≤
We proceed to the first proof.
Proof of Theorem 1: Let ǫ > 0. Let δ > 0 be small enough such that
1
δ min e−14, , and (6)
≤ 2
(cid:26) (cid:27)
1+δ
(1+δ)2 + 2+ǫ. (7)
(1 δ)2 ≤
−
Let ρ(ǫ) > 0 be small enough such that
ρ(ǫ) δ 1 e−2(1δ−2δ) 1 . (8)
≤ 1+δ − 8ln 1
(cid:18) (cid:19) δ
Let ρ (0,ρ(ǫ)). Note that (8) implies ρ δ. Let G be a conn(cid:0)ect(cid:1)ed graph of maximum degree
∈ ≤
at least 1 and girth at least 5.
ρ
Let
1
V = u V(G) : d (u) < , and
1 ∈ G ρ
(cid:26) (cid:27)
V = V(G) V .
2 1
\
Note that φ (u) = 1 for u V , and V = .
ρ 1 2
∈ 6 ∅
Throughout this proof, we write ‘H(X)’ instead of ‘H (X)’.
(G,φρ)
Claim 1 There is a set X V with the following properties.
0 2
⊆
3
(i) N (u) (V H(X )) 1 d (u) for every u V X .
| G ∩ 1 \ 0 | ≤ 1+δ G ∈ 2 \ 0
(ii) X (1+δ)ρn(G).
0
| | ≤
Proof: Let (x ,...,x ) be a maximal sequence of distinct vertices from V such that
1 k 2
1
N (x ) V H( x : 1 j i 1 ) > d (x )
G i 1 j G i
∩ \ { ≤ ≤ − } 1+δ
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
for 1 i k. (cid:12) (cid:12)
≤ ≤
Let X = x ,...,x . The maximality of the sequence (x ,...,x ) implies (i).
0 1 k 1 k
{ }
Since φ (u) = 1 for u V , and d (x ) 1 for 1 i k, we have
ρ ∈ 1 G i ≥ ρ ≤ ≤
H( x : 1 j i ) H( x : 1 j i 1 ) N (x ) V H( x : 1 j i 1 )
j j G i 1 j
| { ≤ ≤ } |−| { ≤ ≤ − } | ≥ ∩ \ { ≤ ≤ − }
(cid:12) 1 (cid:16) (cid:17)(cid:12)
> (cid:12) d (x ) (cid:12)
(cid:12)1+δ G i (cid:12)
1
,
≥ (1+δ)ρ
which implies k < H( x ,...,x ) n(G). It follows that X = k (1 + δ)ρn(G),
(1+δ)ρ | { 1 k} |✷≤ | 0| ≤
which completes the proof of the claim.
Fori N, let X bearandomsubset ofV(G) H(X )that containseach vertex of V(G) H(X )
i 0 0
∈ \ \
independently at random with probability
ρ
p(1) = . (9)
(1 δ)
−
(6)
Notethatρ δ impliesp(1) δ 1. Furthermore, notethatX ischosendeterministically,
≤ ≤ (1−δ) ≤ 0
and that the random sets X ,X ,... are chosen independently of each other.
1 2
Let Y = X , and, for i N, let
0 0
∈
Y = X H(Y ... Y ), and
i i 0 i−1
\ ∪ ∪
Y = Y Y ... Y .
≤i 0 1 i
∪ ∪ ∪
By construction, X X ... X H(Y ) Y H(Y ), which implies
0 1 i ≤i−1 i ≤i
∪ ∪ ∪ ⊆ ∪ ⊆
H(X X ... X ) = H(Y ).
0 1 i ≤i
∪ ∪ ∪
Claim 2 P u H(X X ) δ for i N and u V H(X ).
0 i 2 0
6∈ ∪ ≤ ∈ ∈ \
Proof: Let (cid:2) (cid:3)
N = N (u) (V H(X )),
1 G 1 0
∩ \
N = N (u) (V H(X )), and
2 G 1 0
\ ∪
N = N (u) H(X ).
3 G 0
∩
By Claim 1, N 1 d (u), which implies
| 1| ≤ 1+δ G
δ u∈V2 δ
N + N = d (u) N d (u) . (10)
2 3 G 1 G
| | | | −| | ≥ 1+δ ≥ (1+δ)ρ
4
Let G be the subgraph of G that arises from the induced subgraph
u
G N (u) (N (v) u )
G G
∪ \{ }
" #
v[∈N2
by removing, for every vertex v in N and every vertex w in N (v) u , every edge incident
2 G
\{ }
with w except for the edge vw. Since G has girth at least 5, all vertices in N N are isolated
1 3
∪
in G , and each vertex v in N belongs to a component of G that is a star of order d (v) with
u 2 u G
center v whose set of endvertices is N (v) u .
G
\{ }
Let
H = H (H(X ) X ) V(G ) .
u (Gu,φρ) 0 ∪ i ∩ u
(cid:16) (cid:17)
Since G is a subgraph of G, we have H H(H(X ) X ) = H(X X ).
u u 0 i 0 i
⊆ ∪ ∪
The structure of G and the choice of X implies that
u i
every vertex in N belongs to H with probability p(1), and
1 u
•
every vertex in N belongs to H with probability 1.
3 u
•
Let v N . Note that v H implies v X as well as (N (v) u ) X < ρd (v), that is,
2 u i G i G
∈ 6∈ 6∈ | \{ } ∩ |
(N [v] u ) X < ρd (v), where N [v] = v N (v). Since N [v] u = d (v), the
G i G G G G G
| \{ } ∩ | { }∪ | \{ }|
definitions of p(1) and X imply
i
P v H P (N [v] u ) X < ρd (v)
u G i G
6∈ ≤ | \{ } ∩ |
h i (=9) Ph (N [v] u ) X < (1 δ)ip(1)d (v)
G i G
| \{ } ∩ | −
= P(cid:2)(N [v] u ) X < (1 δ)E (N [v](cid:3) u ) X
G i G i
| \{ } ∩ | − | \{ } ∩ |
h i
< e−δ2E |(NG[v]\{u})∩Xi| /2, (cid:2) (cid:3)
(cid:2) (cid:3)
where the last inequality is a consequence of Chernoff’s inequality.
Since
E (N [v] u ) X = p(1)d (v) (=9) ρdG(v) v∈V2 1 ,
| G \{ } ∩ i| G (1 δ) ≥ (1 δ)
− −
(cid:2) (cid:3)
we obtain
P v H < e−2(1δ−2δ).
u
6∈
This implies that (cid:2) (cid:3)
every vertex in N N belongs to H with probability more than p(2), where
2 3 u
• ∪
p(2) = 1 e−2(1δ−2δ). (11)
−
By the structure of G , the events v H are all independent for v N (u). Let X be a
u u G u
∈ ∈
random subset of N N that contains each vertex of N N independently at random with
2 3 2 3
∪ (cid:2) (cid:3) ∪
probability p(2).
Note that
(10) δp(2) (8) 1
E (N N ) X = p(2)( N + N ) 8ln , (12)
| 2 ∪ 3 ∩ u| | 2| | 3| ≥ (1+δ)ρ ≥ δ
(cid:18) (cid:19)
h i
5
and
ρd (u) (10) ρ(1+δ) (8) 1 (6) 1
G . (13)
p(2)( N + N ) ≤ δp(2) ≤ 8ln 1 ≤ 2
| 2| | 3| δ
Since u H(X X ) implies (N N ) H < ρd (u), the definition of X and Chernoff’s
0 i 2 3 u G (cid:0) (cid:1) u
6∈ ∪ | ∪ ∩ |
inequality imply
P u H(X X ) P (N N ) H < ρd (u)
0 i 2 3 u G
6∈ ∪ ≤ | ∪ ∩ |
h i h i
P (N N ) X < ρd (u)
2 3 u G
≤ | ∪ ∩ |
h ρdi (u)
= P (N N ) X < G p(2)( N + N )
| 2 ∪ 3 ∩ u| p(2)( N + N ) | 2| | 3|
(cid:20) | 2| | 3| (cid:21)
(12),(13) 1
P (N N ) X < E (N N ) X
2 3 u 2 3 u
≤ | ∪ ∩ | 2 | ∪ ∩ |
(cid:20) (cid:21)
h i
< e−81E |(N2∪N3)∩Xu|
(12) (cid:2) (cid:3)
δ.
≤
✷
Claim 3 P u H(Y ) u H(Y ) δ for i N and u V(G) H(X ).
≤i ≤i−1 0
6∈ | 6∈ ≤ ∈ ∈ \
Proof: Reca(cid:2)ll that the set X0 is chosen d(cid:3)eterministically, that is, the only source of randomness
are the sets X ,X ,..., which are chosen independently of each other. This implies that for
1 2
every two not necessarily distinct vertices u′ and u′′ of V(G), the two events u′ H(X X )
0 i
6∈ ∪
and u′′ H(Y ) are independent.
≤i−1
6∈ (cid:2) (cid:3)
First, let u V . Since H(X X ) H(Y ), Claim 2 and the independence observed
2 0 i ≤i
(cid:2) ∈ (cid:3) ∪ ⊆
above imply
P u H(Y ) u H(Y ) P u H(X X ) u H(Y )
≤i ≤i−1 0 i ≤i−1
6∈ | 6∈ ≤ 6∈ ∪ | 6∈
= P u H(X X )
(cid:2) (cid:3) (cid:2) 0 i (cid:3)
6∈ ∪
δ.
(cid:2) (cid:3)
≤
Next, let u V . Let u ...u be a shortest path in G between u = u and some vertex u in
1 0 ℓ 0 ℓ
∈
V . Since φ (u ) = ... = φ (u ) = 1, we obtain that u H(Y ) implies u H(Y ), which
2 ρ 0 ρ ℓ−1 ≤i ℓ ≤i
6∈ 6∈
implies u H(X X ). Now, Claim 2 and the independence observed above imply
ℓ 0 i
6∈ ∪
P u H(Y ) u H(Y ) P u H(Y ) u H(Y )
≤i ≤i−1 ℓ ≤i ≤i−1
6∈ | 6∈ ≤ 6∈ | 6∈
P u H(X X ) u H(Y )
(cid:2) (cid:3) (cid:2) ℓ 0 i ≤(cid:3)i−1
≤ 6∈ ∪ | 6∈
= P u H(X X )
(cid:2) ℓ 0 i (cid:3)
6∈ ∪
δ.
(cid:2) (cid:3)
≤
✷
Note that u H(Y ) implies u H(Y ), and that P u H(Y ) = 0 for u H(X ). By
≤i ≤i−1 ≤i 0
6∈ 6∈ 6∈ ∈
Claim 3, for i N, we obtain, by linearity of expectation,
∈ (cid:2) (cid:3)
E V(G) H(Y ) = P u H(Y )
≤i ≤i
\ 6∈
(cid:2)(cid:12) (cid:12)(cid:3) u∈V(GX)\H(X0) (cid:2) (cid:3)
(cid:12) (cid:12) = P u H(Y ) u H(Y ) P u H(Y )
≤i ≤i−1 ≤i−1
6∈ | 6∈ | 6∈
u∈V(GX)\H(X0) (cid:2) (cid:3) (cid:2) (cid:3)
δ P u H(Y )
≤i−1
≤ 6∈
u∈V(GX)\H(X0) (cid:2) (cid:3)
= δE V(G) H(Y ) , (14)
≤i−1
\
(cid:2)(cid:12) (cid:12)(cid:3)
(cid:12) 6 (cid:12)
which is equivalent to
1
E V(G) H(Y ) E H(Y ) E H(Y ) . (15)
\ ≤i−1 ≤ 1 δ ≤i − ≤i−1
−
(cid:2)(cid:12) (cid:12)(cid:3) (cid:0) (cid:2)(cid:12) (cid:12)(cid:3) (cid:2)(cid:12) (cid:12)(cid:3)(cid:1)
Iteratively applying (14), we obtain
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
E V(G) H(Y ) δin(G).
≤i
\ ≤
Furthermore, by the choice of X(cid:2)(cid:12)and the definiti(cid:12)o(cid:3)n of Y , we have P u Y u H(Y ) =
i i i ≤i−1
(cid:12) (cid:12) ∈ | 6∈
p(1) and P u Y u H(Y ) = 0, which implies
∈ i | ∈ ≤i−1 (cid:2) (cid:3)
(cid:2) E Y = (cid:3) P u Y
i i
| | ∈
(cid:2) (cid:3) u∈V(GX)\H(X0) (cid:2) (cid:3)
= P u Y u H(Y ) P u H(Y )
i ≤i−1 ≤i−1
∈ | 6∈ 6∈
u∈V(GX)\H(X0) (cid:2) (cid:3) (cid:2) (cid:3)
= p(1) P u H(Y )
≤i−1
6∈ |
u∈V(GX)\H(X0) (cid:2) (cid:3)
= p(1)E V(G) H(Y )
≤i−1
\
(15) p(1)(cid:2)(cid:12)(cid:12)E H(Y ) E(cid:12)(cid:12)(cid:3)H(Y ) .
≤i ≤i−1
≤ 1 δ −
−
Therefore, by Claim 1, (cid:0) (cid:2)(cid:12) (cid:12)(cid:3) (cid:2)(cid:12) (cid:12)(cid:3)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
E Y = E Y Y ... Y
≤i 0 1 i
| | | ∪ ∪ ∪ |
X +E Y ... Y
(cid:2) (cid:3) (cid:2)0 1 (cid:3)i
≤ | | | ∪ ∪ |
p(cid:2)(1) i (cid:3)
X + E H(Y ) E H(Y )
≤ | 0| 1 δ ≤j − ≤j−1
− j=1
X(cid:0) (cid:2)(cid:12) (cid:12)(cid:3) (cid:2)(cid:12) (cid:12)(cid:3)(cid:1)
p(1) (cid:12) (cid:12) (cid:12) (cid:12)
X + n(G)
0
≤ | | 1 δ
−
(9) ρ
= X + n(G)
| 0| (1 δ)2
−
ρ
ρ(1+δ)n(G)+ n(G)
≤ (1 δ)2
−
1
= (1+δ)+ ρn(G).
(1 δ)2
(cid:18) − (cid:19)
Let k N be large enough such that δkn(G)+ 1 < 1.
∈ 1+δ
By Markov’s inequality,
P V(G) H(Y ) < 1 Y < (1+δ)E Y
≤k ≤k ≤k
\ ∧ | | | |
1 P V(G) H(Y ) 1 P Y (1+δ)E Y
(cid:2)(cid:0)(cid:12) ≤(cid:12)k (cid:1) (cid:0) ≤k (cid:2) (cid:3)(cid:1)≤(cid:3)k
≥ − \ ≥ − | | ≥ | |
(cid:12) (cid:12)
1
1 δk(cid:2)n(cid:12)(G) (cid:12) (cid:3) (cid:2) (cid:2) (cid:3)(cid:3)
≥ − (cid:12) − 1+δ (cid:12)
> 0.
By the first moment method [4], this implies the existence of a set D of vertices such that
D (1+δ)E Y
≤k
| | ≤ | |
1
(1+δ) (cid:2)(1+δ(cid:3))+ ρn(G)
≤ (1 δ)2
(cid:18) − (cid:19)
(7)
(2+ǫ)ρn(G),
≤
7
and H(D) = V(G), which completes the proof of Theorem 1. ✷
It is conceivable that (1) remains true without the girth condition as well as for arbitrary values
of ρ in 1 ,1 and sufficiently large girth.
∆(G)
We hproceedito the second proof.
Proof of Theorem 2: Let ρbein (0,1]. Suppose that the tree T is a counterexample ofminimum
order, that is, n(T) 1 and h (T) > ρn(T). Let V = u V(T) : d (u) 1 . Note that
≥ ρ φρ 2 ∈ T ≥ ρ
V is a φ -dynamic monopoly of T. If V 1, then hn (T) V ρn(To), which is a
2 ρ | 2| ≤ φρ ≤ | 2| ≤
contradiction. Hence, V 2.
2
| | ≥
Let u in V be chosen such that a largest component K of T u that contains a vertex from
2
−
V has largest possible order. Suppose that T u has a second component K′ distinct from K
2
−
that contains a vertex from V . Let u′ be in V(K′) V . Since T u′ has a component that
2 2
∩ −
contains all vertices in u V(K), we obtain a contradiction to the choice of u. Hence, K is
{ }∪
the only component of T u that contains a vertex from V . Let R = T V(K). Since K and
2
− −
R both contain a vertex from V , we have n(K) 1 and n(R) 1. By the choice of T, the
2 ≥ ρ ≥ ρ
tree K has a φ -dynamic monopoly D with D ρn(K). Let v be the neighbor of u in K.
ρ
| | ≤
Since ρd (v) 1+ρ(d (v) 1) = 1+ρd (v), the set u D is a φ -dynamic monopoly of T
T T K ρ
≤ − { }∪
of order at most D +1 ρn(K)+1 ρn(K)+ρn(R) = ρn(T). This contradiction completes
✷ | | ≤ ≤
the proof.
If ρ (0,1] is such that 1 is an integer, then K shows that the bound in Theorem 2 is
∈ ρ 1,ρ1−1
tight.
References
[1] E. Ackerman, O. Ben-Zwi, G. Wolfovitz, Combinatorial model and bounds for target set
selection, Theoretical Computer Science 411 (2010) 4017-4022.
[2] S.S. Adams, P. Booth, D.S. Troxell, S.L. Zinnen, Modeling the spread of fault in majority-
based network systems: dynamic monopolies in triangular grids, Discrete Applied Mathe-
matics 160 (2012) 1624-1633.
[3] N. Alon, J. Kahn, P.D. Seymour, Large induced degenerate subgraphs, Graphs and Com-
binatorics 3 (1987) 203-211.
[4] N. Alon and J.H. Spencer, The Probabilistic Method, Wiley, New York, 1992.
[5] P. Borowiecki, F. Go¨ring, J. Harant, D. Rautenbach, The potential of greed for indepen-
dence, Journal of Graph Theory 71 (2012) 245-259.
[6] Y. Caro, New results on the independence number, Technical Report, Tel-Aviv University,
1979.
[7] C.C. Centeno, M.C. Dourado, L.D. Penso, D. Rautenbach, J.L. Szwarcfiter, Irreversible
conversion of graphs, Theoretical Computer Science 412 (2011) 3693-3700.
[8] C.C. Centeno, L.D. Penso, D. Rautenbach, V.G.P. de S´a, Geodetic number versus hull
number in P convexity, SIAM Journal on Discrete Mathematics 27 (2013) 717-731.
3
[9] C.C. Centeno, D. Rautenbach, Remarks on Dynamic monopolies with given average thresh-
olds, Discussiones Mathematicae, Graph Theory 35 (2015) 133-140.
8
[10] C.-L. Chang, Triggering cascades on undirected connected graphs, Information Processing
Letters 111 (2011) 973-978.
[11] C.-L. Chang, Y.-D. Lyuu, Triggering cascades on strongly connected directed graphs,
Theoretical Computer Science 593 (2015) 62-69.
[12] N. Chen, Onthe approximability of influence in social networks, SIAM Journal onDiscrete
Mathematics 23 (2009) 1400-1415.
[13] P.A. Dreyer Jr., F.S. Roberts, Irreversible k-threshold processes: graph-theoretical thresh-
old models of the spread of disease and of opinion, Discrete Applied Mathematics 157 (2009)
1615-1627.
[14] F. Go¨ring, J. Harant, D. Rautenbach, I. Schiermeyer, Locally dense independent sets in
regular graphs of large girth, In: Research Trends in Combinatorial Optimization, Springer
Berlin Heidelberg, 2009, 163-183.
[15] K. Khoshkhah, H. Soltani, M. Zaker, On dynamic monopolies of graphs: The average and
strict majority thresholds, Discrete Optimization 9 (2012) 77-83.
[16] J. Lauer and N. Wormald, Large independent sets in regular graphs of large girth, Journal
of Combinatorial Theory, Series B 97 (2007) 999-1009.
[17] R. Motwani, P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.
[18] D. Peleg, Size bounds for dynamic monopolies, Discrete Applied Mathematics 86 (1998)
263-273.
[19] D.Reichman, Newboundsforcontagioussets, DiscreteMathematics312(2012)1812-1814.
[20] V.K. Wei, A lower bound on the stability number of a simple graph, Technical memoran-
dum, TM 81 - 11217 - 9, Bell laboratories, 1981.
[21] M. Zaker, Generalized degeneracy, dynamic monopolies and maximum degenerate sub-
graphs, Discrete Applied Mathematics 161 (2013) 2716-2723.
9