The relationship between $k$-forcing and $k$-power domination

0.37 MB·

by  Daniela Ferrero

The relationship between k-forcing and k-power
domination
Daniela Ferrero∗ Leslie Hogben† Franklin H.J. Kenter‡
Michael Young §
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March 2, 2017
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1 Abstract

Zero forcing and power domination are iterative processes on graphs where an ini-
]
O
tialsetofverticesareobserved,andadditionalverticesbecomeobservedbasedonsome
C rules. In both cases, the goal is to eventually observe the entire graph using the fewest
h. number of initial vertices. Chang et al. introduced k-power domination in [General-
t ized power domination in graphs, Discrete Applied Math. 160 (2012) 1691-1698] as a
a
m generalization of power domination and standard graph domination. Independently,
[ Amos et al. defined k-forcing in [Upper bounds on the k-forcing number of a graph,

Discrete Applied Math. 181 (2015) 1-10] to generalize zero forcing. In this paper, we
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v combine the study of k-forcing and k-power domination, providing a new approach to
6 analyze both processes. We give a relationship between the k-forcing and the k-power
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domination numbers of a graph that bounds one in terms of the other. We also ob-
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8 tain results using the contraction of subgraphs that allow the parallel computation of
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k-forcing and k-power dominating sets.
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0 Keywords k-power domination, k-forcing, subgraph contraction, Sierpin´ski graphs
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AMS subject classification 05C69, 05C50
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:
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X 1 Introduction
r
a
Zeroforcingwasintroducedasaprocesstoobtainanupperboundforthemaximumnullityof
real symmetric matrices whose nonzero pattern of off-diagonal entries is described by a given
graph [2]. The minimum rank problem was motivated by the inverse eigenvalue problem of
a graph. Independently, zero forcing was introduced by mathematical physicists studying
quantum systems [5]. Since its introduction, zero forcing has attracted the attention of a
∗DepartmentofMathematics,TexasStateUniversity,SanMarcos,TX78666,USA([email protected])
†Department of Mathematics, Iowa State University, Ames, IA 50011, USA ([email protected]) and
American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA ([email protected]).
‡Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, USA
([email protected]).
§Department of Mathematics, Iowa State University, Ames, IA 50011, USA ([email protected])
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large number of researchers who find the concept useful to model processes in a broad range
of disciplines. The need for a uniform framework for the analysis of the diverse processes
where the notion of zero forcing appears led to the introduction of a generalization of zero
forcing called k-forcing [3].
Amos et al. proposed k-forcing in [3] as the following graph coloring game. Assume
the vertices of a graph are colored in two colors, say white and blue. Iteratively apply
the following color change rule: if u is a blue vertex with at most k white neighbors, then
change the color of all the neighbors of u to blue. Once this rule does not change the color
of any vertex, if all vertices are blue, the original set of blue vertices is a k-forcing set of
G. The original zero forcing is 1-forcing under this definition. Because the problem of
deciding whether a graph admits a 1-forcing set of a given maximum size is NP-complete
even if restricted to planar graphs [1, Theorem 2.3.1], the general problem of finding forcing
sets cannot be solved algorithmically for large graphs without the development of further
theoretical tools.
Power domination was introduced by Haynes et al. in [9] when using graph models
to study the monitoring process of electrical power networks. When a power network is
modeled by a graph, a power dominating set provides the locations where monitoring devices
(Phase Measurement Units, or PMUs for short) can be placed in order to monitor the power
network. Finding optimal PMU placements is an important practical problem in electrical
engineering due to the cost of PMUs and network size. Although power domination is
substantially different from standard graph domination, the notion of k-power domination
was proposed as a generalization of both power domination (k = 1) and standard graph
domination (k = 0) [6].
Chang et al. defined k-power domination in [6] using sets of observed vertices. Given a
graph G and a set of vertices S, initially all vertices in S and their neighbors are observed;
all other vertices are unobserved. Iteratively apply the following propagation rule: if there
exists an observed vertex u that has k or fewer unobserved neighbors, then all the neighbors
of u are observed. Once this rule does not produce any additional observed vertices, if
all vertices of G are observed, S is a k-power dominating set of G. Many problems outside
graphtheorycanbeformulatedintermsofminimumk-powerdominatingsets[6]somethods
to obtain them are highly desired. An algorithmic approach has been attempted, but the
problem of deciding if a graphs admits a k-power dominating set of a given maximum size
is NP-complete [6].
Although k-forcing and k-power domination have been studied independently, an in-
depth analysis of k-power domination leads to the study of k-forcing. Indeed, after the
initial step in which a set observes itself and its neighbors, the observation process in k-
power domination proceeds exactly as the color changing process in k-forcing. The aim of
this paper is to establish a precise connection between k-forcing and k-power domination to
facilitate the transference of results, proofs, and methods between them, and ultimately to
advance research on both problems.
Throughout this paper we work on k-forcing and k-power domination concurrently,
using results in one process as stepping stones for results in the other one. In Section 2 we
present the definitions and notation that we use in the rest of the paper. In Section 3 we
give some core results and remarks that we use in the sections that follow.
In Section 4 we examine the effect of subgraph contraction in k-power domination and
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k-forcing. We obtain upper and lower bounds for the change in the k-power domination
number produced by the contraction of a subgraph. Note that the contraction of a subgraph
can increase or decrease its k-power domination number. In particular, we prove that the
contraction of subgraphs of small degree can change the k-power domination number by at
mostone. Inthissectionwealsoproposeawaytodecomposeagraphinordertobounditsk-
powerdominationnumberintermsofthatofsmallersubgraphs. Thiscanallowcomputation
ofk-powerdominatingsetstoruninparallel. Wealsogivetheanalogousresultsfork-forcing.
In Section 5 we present a lower bound for the k-power domination number of a graph
in terms of its k-forcing number. This bound generalizes a known result for k = 1 that gives
the only lower bound for the power domination number of an arbitrary graph available so
far [4]. As an application, we find an upper bound for the k-forcing number of a graph in
terms of its maximum degree.
2 Definitions and notation
A graph is an ordered pair G = (V,E) where V = V(G) is a finite nonempty set of vertices
and E = E(G) is a set of unordered pairs of distinct vertices called edges (i.e., in this work
graphs are simple and undirected). The order of G is |G| := |V(G)|. Two vertices u and
v are adjacent or neighbors in G if {u,v} ∈ E(G). The (open) neighborhood of a vertex
v is the set N (v) = {u ∈ V : {u,v} ∈ E}, and the closed neighborhood of v is the set
G
N [v] = N (v)∪{v}. Similarly, for any set of vertices S, N (S) = ∪ N (v) and N [S] =
G G G v∈S G G
∪ N [v]. The degree of a vertex v is deg (v) := |N(v)|. The maximum and minimum
v∈S G G
degree of G are ∆(G) := max{deg (v) : v ∈ V} and δ(G) := min{deg (v) : v ∈ V},
G G
respectively; a graph G is regular if δ(G) = ∆(G). We will omit the subscript G when the
graph G is clear from the context.
A path joining u,v ∈ V is a sequence of vertices u = x ,x ,...,x = v such that
0 1 r
{x ,x } ∈ E for each i = 0,...,r − 1. A graph G is connected if there is a path joining
i i+1
every pair of different vertices. If a graph is not connected, each maximal connected sub-
graph is a component of G. In this paper, c(G) denotes the number of components of G
and G ,...,G denote the components of G. Most of the results in this work are given
1 c(G)
for connected graphs, since if a graph is not connected, we can apply the results to each
component.
If X is a set of vertices of G, the subgraph induced by X (in G) is denoted as G[X];
it has vertex set X and edge set {{u,v} ∈ E : u,v ∈ X}. The graph G−X is defined as
G[V \X]. The contraction of X in G is the graph G/X obtained by adding a vertex v to
X
G−X with N (v ) = N [X]\X. Note that G/X does not require G[X] to be connected
G/X X G
whereas the standard use of graph contraction does.
In a graph G = (V,E), consider an arbitrary coloring of its vertices in two colors,
say blue and white, and let T denote the set of blue vertices. The color changing process
in k-forcing can be formally described by associating to T the family of sets (Fi (T))
G,k i≥0
recursively defined by the following rules.
1. F0 (T) = T,
G,k
2. Fi+1(T) = Fi (T)∪{N(v) : v ∈ Fi (T) and 1 ≤ |N (v)\Fi (T)| ≤ k}, for i ≥ 0.
G,k G,k G,k G G,k
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A set T ⊆ V is a k-forcing set of G if there is an integer t such that Ft (T) = V. A
G,k
minimum k-forcing set is a k-forcing set of minimum cardinality. The k-forcing number of G
is the cardinality of a minimum k-forcing set and is denoted by Z (G). If v ∈ Fi (T) and
k G,k
|N(v)\Fi (T)| ≤ k then v is said to k-force (or simply force if k is clear from the context)
G,k
every vertex in N(v)\Fi (T).
G,k
Let k be a nonnegative integer. The definition of k-power domination on a graph G
will be given in terms of a family of sets, (Pi (S)) , associated to each set of vertices S
G,k i≥0
in G.
1. P0 (S) = N[S],
G,k
2. Pi+1(S) = Pi (S)∪{N(v) : v ∈ Pi (S) and 1 ≤ |N (v)\Pi (S)| ≤ k}, for i ≥ 0.
G,k G,k G,k G G,k
A set S ⊆ V is a k-power dominating set of G if there is an integer (cid:96) such that
P(cid:96) (S) = V. A minimum k-power dominating set is a k-power dominating set of minimum
G,k
cardinality. The k-power domination number of G is the cardinality of a minimum k-power
dominating set and is denoted by γ (G).
P,k
Next we recall the definition of standard graph domination. A vertex v dominates all
vertices in N [v]. A set S ⊆ V is a dominating set of G if N [S] = V. The minimum
G G
cardinality of a dominating set is the domination number of G, denoted by γ(G).
Note that 1-forcing coincides with zero forcing [3], while 1-power domination is exactly
power domination and 0-power domination coincides with domination [6].
3 Preliminaries
The following observations follow directly from the definitions of k-power domination and
k-forcing, and provide the initial connection between both concepts.
Observation 3.1. In any graph G, if T is a k-forcing set, all sets (Fi (T)) are k-forcing
G,k i≥0
sets of G; if S is a k-power dominating of G, the sets (Pi (S)) are also k-forcing sets
G,k i≥0
of G.
Observation 3.2. In any graph G, if T is a k-forcing set of G then T is also a k-power
dominating set. The converse is not necessarily true, but S is a k-power dominating set if
and only if N[S] is a k-forcing set. As a consequence, γ (G) ≤ Z (G) ≤ γ (G)(∆(G)+1).
P,k k P,k
Observation 3.3. In a graph G, S (cid:40) V(G) is a k-power dominating set of G if and only if
N[S]\S is a k-forcing set of G−S.
Note that given a graph G = (V,E) and S ⊆ X ⊆ V, it is possible that for some
x ∈ X, deg (x) < deg (x). Therefore, the k-power domination process starting with
G[X] G
S in G is different from the one starting with S in G[X]. As a consequence, S being a
k-power dominating set of G[X] does not imply that S can k-observe all vertices in X when
propagating in G. Analogously, if T ⊆ X ⊆ V then T being a k-forcing set of G[X] does
not imply that T can k-force X in G. This observation motivates the following definitions.
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Definition 3.4. Let G = (V,E) be a graph and let A ⊆ X ⊆ V. We say that A is a
k-forcing set of X in G if there exists a nonnegative integer t such that X ⊆ Ft (A).
G,k
Definition 3.5. Let G = (V,E) be a graph and let A ⊆ X ⊆ V. We say that A is a k-power
dominating set of X in G if there exists a nonnegative integer (cid:96) such that X ⊆ P(cid:96) (A).
G,k
The proofs of the next results are straightforward, and are omitted.
Lemma 3.6. Let T be a k-forcing set of a graph G. Let A ⊆ T.
1) If A is k-forcing set of T in G, then A is a k-forcing set of G;
2) If A is k-power dominating set of T in G, then A is a k-power dominating set of G.
Lemma 3.7. Let S be a k-power dominating set of a graph G. Let A ⊆ S.
1) If A is k-forcing set of N[S] in G, then A is a k-forcing set of G;
2) If A is k-power dominating set of N[S] in G, then A is a k-power dominating set of
G.
Lemma 3.8. Let G = (V,E) be a graph and X ⊆ V such that G[X] is connected and
deg (x) ≤ k + 1 for every x ∈ X. Let u be an arbitrary vertex in X. Then {u} is a
G
(minimum) k-power dominating set of N[X] in G. In addition, if deg (u) ≤ k, then {u} is
G
also a (minimum) k-forcing set of N[X] in G.
Proof. If S = {u}, then P0 (S) = N[u] and F0 (S) = {u}. Since deg (x) ≤ k + 1
G,k G,k G
for every x ∈ X, x (cid:54)= u has at most k unobserved neighbors when x is observed. Thus,
Pi (S) = N[Pi−1(S)] for every integer i ≥ 1. Since G[X] is connected, there exists an
G,k G,k
integer r ≥ 1 such that X ⊆ Pr (S). Once all vertices in X are observed, each of them
G,k
can have at most k unobserved neighbors, so such a vertex can observe any unobserved
neighbors. Thus, N[X] ⊆ Pr+1(S) and S is a k-power dominating set of N[X] in G. Now
G,k
suppose deg (u) ≤ k. Then F1 (S) = N[S] and the argument proceeds as before.
G G,k
The following result follows immediately from Lemma 3.8, but is already known for
k-power domination [6, Lemma 7]; a slightly weaker version for k-forcing is given in [3,
Proposition 2.3].
Corollary 3.9. Let G be a connected graph. If ∆(G) ≤ k + 1, then γ (G) = 1; if in
P,k
addition δ(G) ≤ k, then Z (G) = 1.
k
WhenG[X]isnotconnected, weapplyLemma3.8ineachofitscomponentsandobtain
the following result.
Corollary 3.10. Let G = (V,E) be a connected graph, X ⊆ V and u ∈ V(G[X] ) for every
j j
j = 1,...,c(G[X]). Let S = {u ,...,u }. If deg (x) ≤ k +1 for every x ∈ X, then S
1 c(G[X]) G
is a minimum k-power dominating set of N[X] in G; if in addition deg (u ) ≤ k for every
G j
j = 1,...,c(G[X]), then S is a minimum k-forcing set of N[X] in G.
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Proof. By Lemma 3.8, for every j = 1,...,c(G[X]), {u } is a k-power dominating set of
j
G[X] in G. Thus, S is a k-power dominating set of N[X] in G. Since every k-power
j
dominating set of N[X] must have at least one vertex in each component of G[X] and
|S| = c(G[X]) we conclude that S is a minimum k-power dominating set of N[X] in G. The
argument for k-forcing is analogous.
Lemma 3.11. Let G = (V ,E ) and H = (V ,E ) be two graphs. Let A ⊆ V and B ⊆ V
G G H H G H
such that (i) G−A = H −B and (ii) N [A]\A = N [B]\B. Then
G H
1) A is a k-power dominating set of G if and only if B is a k-power dominating set of H;
2) N [A] is a k-forcing set of G if and only if N [B] is a k-forcing set of H.
G H
Proof. 1) If A is a k-power dominating set of G, then N [A]\A is a k-forcing set of G−A
G
by Observation 3.3. Since G − A = H − B and N [A] \ A = N [B] \ B, we substitute
G H
N [A]\A and G−A with N [B]\B and H −B, respectively, and obtain that N [B]\B
G H H
is a k-forcing set of H −B. By Observation 3.3, B is a k-power dominating set of H.
2) If N [A] is a k-forcing set of G, then A is a k-power dominating set of G by
G
Observation 3.2. Using 1) we conclude that B is a k-power dominating set of H, and by
Observation 3.2 we conclude that N [B] is a k-forcing set of H.
H
Corollary 3.12. Let G = (V ,E ) and H = (V ,E ) be two graphs. Let A ⊆ V and
G G H H G
B ⊆ V such that (i) G−A = H−B and (ii) N [A]\A = N [B]\B. Let P ⊆ V \A = V \B.
H G H G H
Then
1) A∪P is a k-power dominating set of G if and only if B ∪P is a k-power dominating
set of H;
2) N [A]∪P is a k-forcing set of G if and only if N [B]∪P is a k-forcing set of H.
G G
Proof. Define A(cid:48) = A ∪ P and B(cid:48) = B ∪ P. Then G − A(cid:48) = H − B(cid:48) and N [A(cid:48)] \ A(cid:48) =
G
N [B(cid:48)]\B(cid:48), so we can apply Lemma 3.11 with G, H, A(cid:48) and B(cid:48).
H
While all the previous results include analogous statements for k-forcing and a k-power
domination, the following lemma does not have a k-forcing analog.
Lemma 3.13. [6, Lemma 9] If G is connected and ∆(G) ≥ k + 2, then there exists a
minimum k-power dominating set S such that deg(v) ≥ k +2 for all v ∈ S.
To see that there is no k-forcing analog to Lemma 3.13 it is sufficient to consider K ,
1,n
the complete bipartite graph with one vertex in one part and n vertices in the other. As
shown in [3], if n > k every minimum k-forcing set contains at least one vertex of degree 1.
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4 Graph contraction
Definition 4.1. Let G be a graph and let X ⊆ V(G). Define X(cid:98) to be the graph obtained
from G[X] by attaching to each one of its vertices as many pendent vertices as its number
of neighbors in G−X.
Lemma 4.2. Let G be a connected graph and let X ⊆ V(G). There exists S ⊆ X such that
S is a minimum k-power dominating set of X(cid:98).
Proof. Supposefirstthat∆(X(cid:98)) ≤ k+1. ThenbyLemma3.8anyonevertexofX isak-power
dominating set for N[X] = X(cid:98); a one vertex k-power dominating set is necessarily minimum.
Now assume ∆(X(cid:98)) ≥ k+2. By definition of X(cid:98), deg (u) = 1 for every u ∈ V(X(cid:98))\X. Since
X(cid:98)
∆(X(cid:98)) ≥ k+2, by Lemma 3.13 there exists a minimum k-power dominating set S of X(cid:98) that
contains only vertices in X.
For the same reasons why there is no k-forcing analog to Lemma 3.13, there is no
k-forcing analog to Lemma 4.2. Indeed, if x ∈ V(G) and deg (x) ≥ k + 1, a minimum
G
k-forcing set of {(cid:100)x} must contain a vertex of degree 1.
Lemma 4.3. Let G be a connected graph and let X ⊆ V(G). If S ⊆ X is a minimum
k-power dominating set of X(cid:98), then S is a k-power dominating set of N [X] in G.
G
Proof. Each vertex in V(X(cid:98))\X arises from a vertex y ∈/ X that is a neighbor of a vertex
x ∈ X. For every x ∈ X, let N denote the (possibly empty) set of neighbors of x in
x
V(X(cid:98))\X (i.e., N = N (x)\X) and let N(cid:48) = N (x)\X. Since S ⊆ X and deg (u) = 1
x X(cid:98) x G X(cid:98)
for every u ∈ V(X(cid:98)) \ X, none of the vertices in N can be observed before x is observed,
x
and moreover, all vertices in N are observed simultaneously. Since for every x ∈ V(X(cid:98)),
x
deg (x) = deg (x), the only difference between the k-power domination process starting
X(cid:98) G
with S in X(cid:98) and the one starting with S in G is that when the vertices in N are observed
x
in X(cid:98), the unobserved vertices in N(cid:48) become observed in G. The reason why some vertices
x
in N(cid:48) could have been observed earlier is that a vertex in G−X could have more than one
x
neighbor in X so (N(cid:48)) are not necessarily disjoint. Since for every w ∈ N [X]\X there
x x∈X G
exists x ∈ X such that w ∈ N(cid:48), all vertices in N [X] are observed.
x G
Theorem 4.4. Let G = (V,E) be a connected graph. If X ⊆ V,
γ (G/X)−1 ≤ γ (G) ≤ γ (G/X)+γ (X(cid:98))
P,k P,k P,k P,k
and both bounds are tight.
Proof. Let H = G/X. By Lemma 4.2 there exists P(cid:98) ⊆ X such that P(cid:98) is a minimum k-power
dominating set of X(cid:98) and by Lemma 4.3, P(cid:98) is also a k-power dominating set of N [X] in G.
G
To prove the upper bound we show that if P is a k-power dominating set of H, then
(P \{v })∪P(cid:98) is a k-power dominating set of G.1 Since P(cid:98) is a k-power dominating set of
X
1Note that whether vX ∈/ P or vX ∈P does not affect the conclusion, since in any case |S|≤|P|+|P(cid:98)|=
γP,k(H)+γP,k(X(cid:98)); we only exclude vX from S to guarantee S ⊆V(G).
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N [X] in G, clearly (P \{v })∪P(cid:98) is a k-power dominating set of N [P \{v }]∪N [X] =
G X G X G
N [P \{v }∪X] in G. We will prove that (P \{v })∪X is a k-power dominating set of G,
G X X
whichbyLemma3.7sufficestoconcludethat(P\{v })∪P(cid:98) isak-powerdominatingsetofG.
X
Let A = X and B = {v }. Since H = G/X, G−A = H−B and N [A]\A = N [B]\B, we
X G H
apply Corollary 3.12 and conclude that (P\{v })∪A is a k-power dominating set of G if and
X
only if (P\{v })∪B is a k-power dominating set of H. Since B = {v }, (P\{v })∪B = P
X X X
and P is a k-power dominating set of H, (P \ {v }) ∪ A = (P \ {v }) ∪ X is a k-power
X X
dominating set of G.
To prove the lower bound, we show that if S is a minimum k-power dominating set
of G, then (S \ X) ∪ {v } is a k-power dominating set of H. As above, let A = X and
X
B = {v } so G−A = H −B and N [A]\A = N [B]\B. Then we apply Corollary 3.12
X G H
to conclude that (S \ X) ∪ A is a k-power dominating set of G if and only if (S \ X) ∪ B
is a k-power dominating set of H. Since X = A, then (S \X)∪A = S and it is a k-power
dominating set of G. Then (S\X)∪B = (S\X)∪{v } is a k-power dominating set of H.
X
Thus, γ (G/X) ≤ |(S \X)∪{v }| ≤ |S|+1 = γ (G)+1.
P,k X P,k
To prove the upper bound is tight, for each integer q ≥ k we define a graph U and
q
a set X ⊆ V(U ) such that γ (U ) = γ (U /X)+γ (X(cid:98)) (see Figure 1). Consider two
q P,k q P,k q P,k
disjoint copies of K , say G and G(cid:48), and vertices x ∈ V(G) and y ∈ V(G(cid:48)). Construct U
q+2 q
by adding the edge e = {x,y} and define X = V(G(cid:48))\{y}. Then γ (U ) = 2, γ (X(cid:98)) = 1,
P,k q P,k
and γ (U /X) = 1.
P,k q
X
v
X
U X(cid:98) U /X
2 2
Figure 1: The graphs U , X(cid:98), and U /X defined in Theorem 4.4 are shown. In each case, a
2 2
minimum 2-power dominating set is indicated by coloring.
X
v
X
L L /X
2 2
Figure 2: The graphs L and L /X defined in Theorem 4.4 are shown. In each case, a
2 2
minimum 2-power dominating set is indicated by coloring.
To show the lower bound is tight, for each integer q ≥ k we define a graph L and a
q
set X ⊆ V(L ) such that γ (L /X)−1 = γ (L ) (see Figure 2). Assume first that k ≥ 2.
q P,k q P,k q
Construct L starting with a cycle of length 2q with vertices v ,...,v . Attach a pendent
q 1 2q
vertex to each vertex v , for i = 1,...,2q. Then attach q+1 pendent vertices to the pendent
i
8
neighbor of v , so γ (L ) = 1. For X = {v ,...,v }, γ (L /X) = 2. Now suppose k = 1,
1 P,k q 1 2p P,k q
and begin with a path of length 6 with vertices v ,...,v . Construct L by attaching q
0 6 q
pendent vertices to v . If X = {v ,v ,v }, then γ (L ) = 1 and γ (L /X) = 2.
0 1 3 5 P,1 q P,1 q
The next example shows that it is possible to find a graph G and a subgraph X for
which the gap between γ (G) and γ (G/X) is arbitrarily large.
P,k P,k
Example 4.5. Given a positive integer c, define T as the tree obtained by adding k +2
k,c
leaves to each leaf of K . If X is the set of all vertices of degree greater than one in T ,
1,c k,c
then γ (T ) = γ (X(cid:98)) = c and γ (T /X) = 1.
P,k k,c P,k P,k k,c
Corollary 4.6. Let G = (V,E) be a connected graph. Let X ⊆ V such that deg (x) ≤ k+1
G
for every x ∈ X. Then
γ (G/X)−1 ≤ γ (G) ≤ γ (G/X)+c(G[X]).
P,k P,k P,k
Proof. It is sufficient to show that γ (X(cid:98)) ≤ c(G[X]). Observe that deg (x) ≤ k + 1 for
P,k G
every x ∈ X implies that deg (x) ≤ k + 1 for every x ∈ V(X(cid:98)). By Corollary 3.10 there
X(cid:98)
exists a k-power dominating set of N [X] = V(X(cid:98)) in G with cardinality c(X(cid:98)) = c(G[X]),
X(cid:98)
so γ (X(cid:98)) ≤ c(G[X]).
P,k
Corollary 4.7. Let G = (V,E) be a connected graph. Let X ⊆ V such that G[X] is connected
and deg (x) ≤ k +1 for every x ∈ X.Then
G
γ (G/X)−1 ≤ γ (G) ≤ γ (G/X)+1.
P,k P,k P,k
Proposition 4.8. Let G = (V,E) be a connected graph. Let X ⊆ V such that G[X] is
connected and deg (x) ≤ 2 for every x ∈ X. Then γ (G/X) ≤ γ (G) and this bound it
G P,1 P,1
tight.
Proof. Since ∆(G) ≤ 2 implies that G itself is a path or a cycle, without loss of generality
we can assume ∆(G) ≥ 3. By Lemma 3.13, there exists a minimum k-power dominating set
S of G such that deg (u) ≥ 3 for every u ∈ S, so S ⊆ V \ X. We prove that S is also a
G
k-power dominating set of H.
As in the proof of Theorem 4.4, S ∪{v } is a k-power dominating set of H. Then by
X
Observation 3.2, N [S ∪{v }] = N [S]∪N [v ] is a k-forcing set of H.
H X H H X
Note that S ⊆ V \X implies P0 (S)\X = P0 (S)\{v } and as long as Pi (S) ⊆
G,k H,k X G,k
V \ X, Pi (S) = Pi (S). Let x be a vertex of X that is observed first (meaning that
G,k H,k
no vertex of X has been observed earlier), and let y be the vertex in G−X that dominates
or forces x at time t (x ∈ P0 (S) or x ∈ Pt (S)\Pt−1(S) for t ≥ 1). Since deg (y) ≥
G,k G,k G,k G
deg (y), y can also dominate or force in H. Thus v ∈ Pt (S). Since deg (v ) ≤ 2, it
H X H,k H X
takes at most one additional application of k-forcing to observe all vertices in N [v ], so
H X
N [v ] ⊆ Pt+1(S).
H X H,k
Since N [S] = P0 (S) ⊆ Pt+1(S) and N [v ] ⊆ Pt+1(S), then N [S ∪ {v }] =
H H,k H,k H X H,k H X
N [S] ∪ N [v ] ⊆ Pt+1(S). Moreover, since N [S ∪ {v }] is a k-forcing set of H, so is
H H X H,k H X
Pt+1(S) and therefore, S is a k-power dominating set of H.
H,k
To prove the tightness, observe that for n ≥ 3, contracting the set of all vertices of
degree 2 in the path P of order n produces the path P . Now, γ (P ) = γ (P ) = 1.
n 3 P,1 n P,1 3
9
Due to the computational complexity of the k-power domination problem, efficient
algorithms to approximate of optimal k-power dominating sets are of practical importance.
Theorem 4.4 could help in the parallel search for k-power dominating sets. The following
result provides a theoretical framework to study practical uses of graph decomposition as a
tool for the parallel computation of k-power dominating sets.
Theorem 4.9. Let G = (V,E) be a connected graph and let P ,...,P be a partition of V.
1 r
Then
r
(cid:88)
γ (G) ≤ γ (P(cid:98)).
P,k P,k i
i=1
Proof. By Lemma 4.2, for every i = 1,...,r there exists S ⊆ P such that S is a minimum
i i i
k-power dominating set of P(cid:98). By Lemma 4.3, S is also a k-power dominating set of N [P ]
i i G i
in G, and as a consequence, S = ∪r S is a k-power dominating set of G. Then γ (G) ≤
i=1 i P,k
|S| ≤ (cid:80)r |S | = (cid:80)r γ (P(cid:98)).
i=1 i i=1 P,k i
To prove that the bound in Theorem 4.9 is tight we will use the family of Sierpin´ski
graphs whose definition we recall, using the notation in [8]. Given two positive integers n
and p the Sierpin´ski graph Sn has as vertices all n-tuples of integers in {0,1,...,p − 1}
p
denoted as s s ···s s . Two vertices s ···s and t ···t are adjacent in Sn if and only
n n−1 2 1 n 1 n 1 p
if there exists an r with 1 ≤ r ≤ n such that
i) s = t for every i ∈ {r+1,...,n},
i i
ii) s (cid:54)= t , and
r r
iii) s = t and t = s for every i ∈ {1,...,r−1}
i r i r
The definition of Sierpin´ski graphs implies that S1 = K and if n ≥ 2, Sn has pn−i
p p p
induced copies of Si. Moreover, the vertices in each of those copies coincide in the n − i
p
leftmost digits s ···s [8]. If s is a (n−i)-tuple of integers in {0,...,p−1}, let sSi denote
n i+1 p
the set of vertices of Sn whose leftmost n−i digits coincide with s. For simplicity, we use
p
Sn[s] to denote Sn[sSi] (the subgraph induced by sSi in Sn), so Sn[s] is isomorphic to Si.
p p p p p p p
Lemma 4.10. Given integers n ≥ 4, k ≥ 1 and p ≥ k+2, let s be a (n−3)-tuple of integers
in {0,...,p−1}. Then γ (s(cid:100)S3) = γ (S3).
P,k p P,k p
Proof. Fix k ≥ 1, p ≥ k + 2, and n ≥ 4. We begin by determining the degree of a vertex
sxyz in Sn and in Sn[s] ∼= S3. The definition of the Sierpin´ski graph Sn implies that vertices
p p p p
of the form an have degree p − 1 and all the other vertices have degree p in Sn. If xyz is
p
nonconstant, then vertex sxyz has degree p in both Sn and Sn[s], so sxyz does not have
p p
any pendent vertices added in s(cid:100)S3. Now consider the constant sequence aaa. If s (cid:54)= an−3,
p
then vertex saaa has degree p in Sn but degree p−1 in Sn[s], so one leaf is added to saaa
p p
in s(cid:100)S3. If s = an−3, then vertex saaa = an has degree p−1 in both Sn and Sn[s], so saaa is
p p p
unchanged in s(cid:100)S3.
p
10