PREPRINT
國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
~ mathlib/preprint/2012- 08.pdf
Algorithmic aspects of k-domination in graphs
James K. Lan and Gerard Jennhwa Chang
September 24, 2012
Algorithmic aspects of k-domination in graphs
IJames K. Lana,∗, Gerard Jennhwa Changa,b,c
aDepartment of Mathematics, National Taiwan University, Taipei 10617, Taiwan
bTaida Institute for Mathematical Sciences, National Taiwan U., Taipei 10617, Taiwan
cNational Center for Theoretical Sciences, Taipei Office, Taiwan
Abstract
For a positive integer k, a k-dominating set of a graph G is a subset D ⊆ V (G) such that every vertex not in D is adjacent to at least k vertex in D.
The k-domination problem is to determine a minimum k-dominating set of G. This paper studies k-domination problem in graphs from an algorithmic point of view. In particular, we present a linear-time algorithm for the k- domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph. This class of graphs include trees, block graphs, cacti and block-cactus graphs. We also establish NP-completeness of the k-domination problem in split graphs.
Keywords: k-Domination, Tree, Block graph, Cactus, Block-cactus graph, Split graph, Algorithm, NP-complete
1. Introduction
All graphs in this paper are simple, i.e., finite, undirected, loopless and without multiple edges. Domination is a core NP-complete problem in graph theory and combinatorial optimization. It has many applications in the real world such as location problems, sets of representatives, social network the- ory, etc; see [3, 12] for more interesting applications. A vertex is said to dominate itself and all of its neighbors. A dominating set of a graph G is a
IThis research was partially supported by the National Science Council of the Republic of China under grants NSC100-2811-M-002-146 and NSC98-2115-M-002-013-MY3.
∗Corresponding author.
Email addresses: drjamesblue@gmail.com (James K. Lan), gjchang@math.ntu.edu.tw (Gerard Jennhwa Chang)
subset D of V (G) such that every vertex not in D is dominated by at least one vertex in D. The domination number γ(G) of G is the minimum size of a dominating set of G. The domination problem is to find a minimum dominating set of a graph.
It is well-known that given any minimum dominating set D of a graph G, one can always remove two edges from G such that D is no longer a dominating set for G [12, p.184]. The idea of dominating each vertex multiple times is naturally considered. One of such generalization is the concept of k-domination, introduced by Fink and Jacobson in 1985 [10]. For a positive integer k, a k-dominating set of a graph G is a subset D ⊆ V (G) such that every vertex not in D is dominated by at least k vertices in D. The k- domination number γk(G) of G is the minimum size of a k-dominating set of G. The k-domination problem is to determine a minimum k-dominating set of a graph. The special case when k = 1 is the ordinary domination.
Many of the k-domination results in the literature focused on finding bounds on the number γk(G). In particular, bounds in terms of order, size, minimum degree, maximum degree, domination number, independence num- ber, k-independence number, and matching number were extensively studied [2, 4, 6, 7, 9, 10, 11, 16]; also see the recent survey paper [5].
On the complexity side of the k-domination problem, Jacobson and Peters showed that the k-domination problem is NP-complete for general graphs [14]
and gave linear-time algorithms to compute the k-domination number of trees and series-parallel graphs [14]. The k-domination problem remains NP- complete in bipartite graphs or chordal graphs [1]. More complexity results for the k-domination problem are desirable.
In this paper, we explore efficient algorithms for the k-domination prob- lem in graphs. In particular, we present a linear-time algorithm for the k-domination problem in graphs in which each block is a clique, a cycle or a complete bipartite graph. This class of graphs include trees, block graphs, cacti and block-cactus graphs. We also show that the k-domination problem remains NP-complete in split graphs, a subclass of chordal graphs.
2. Preliminaries
Let G = (V, E) be a graph with vertex set V and edge set E. For a vertex v, the open neighborhood is the set N (v) = { u ∈ V : uv ∈ E } and the closed neighborhood is N [v] = N (v)∪ { v }. The degree deg(v) of a vertex v in G is the number of edges incident to v.
The subgraph of G induced by S ⊆ V is the graph G[S] with vertex set S and edge set { uv ∈ E : u, v ∈ S }. In a graph G = (V, E), the deletion of S ⊆ V from G, denoted by G − S, is the graph G[V \ S]. For a vertex v in G, we write G− v for G − { v }.
In a graph, a stable set (or independent set ) is a set of pairwise nonadja- cent vertices, and a clique is a set of pairwise adjacent vertices. A forest is a graph without cycles. A tree is a connected forest. A leaf of a graph is a vertex with degree one. A vertex v is a cut-vertex if the number of connected components is increased after removing v. A block of a graph is a maximal connected subgraph without any cut-vertex. An end-block of a graph is a block containing at most one cut-vertex. A block graph is a graph whose blocks are cliques. A cactus is a connected graph whose blocks are either an edge or a cycle. A block-cactus graph is a graph whose blocks are cliques or cycles.
3. Labeling method for k-domination
Labeling techniques are widely used in the literatures for solving the domination problem and its variants [3, 8, 13, 15]. For k-domination, we employe the following labeling method which is similar to that in [15]. Given a graph G, a k-dom assignment is a mapping L that assigns each vertex v in G a two-tuple label L(v) = (L1(v), L2(v)), where L1(v)∈ { B, R }, and L2(v) is a nonnegative integer. Here a vertex v with L1(v) = R is called a required vertex ; a vertex v with L1(v) = B is called a bound vertex. An L-dominating set of G is a subset D ⊆ V (G) such that
• if L1(v) = R, then v ∈ D, and
• if L1(v) = B, then either v ∈ D or |N(v) ∩ D| ≥ L2(v).
That is, D contains all required vertices, and for each bound vertex v not in D, v is adjacent to at least L2(v) vertices in D. The L-domination number γL(G) is the minimum size of an L-dominating set in G, such set is called a γL-set of G. Notice that if L(v) = (B, k) for all v ∈ V (G), then γL(G) = γk(G). Thus an algorithm for γL(G) gives γk(G).
Lemma 1. Suppose G is a graph with a k-dom assignment L = (L1, L2).
For a vertex v in G, let G′ = G− v and let L′ be the restriction of L on V (G′) with the modification that L′2(u) = max{ L2(u)− 1, 0 } for u ∈ N(v).
If L1(v) = R or L2(v) > deg(v), then γL(G) = γL′(G′) + 1.
Proof. Suppose D′ is a γL′-set of G′. Set D = D′ ∪ { v }. Since L′ is the restriction of L on V (G′) with the modification on L′2(u) and L2(u)≤ L′2(u)+
1 for u ∈ N(v), D is clearly an L-dominating set of G. Thus γL(G) ≤ |D| =
|D′| + 1 = γL′(G′) + 1.
Conversely, suppose D is a γL-set of G. By the assumption that L1(v) = R or L2(v) > deg(v), v must be included in D. Set D′ = D\ { v }. As L′ is the restriction of L on G′ with the modification on L′2(u) for u ∈ N(v), D′ is an L′-dominating set of V (G′). Hence γL′(G′) + 1≤ |D′|+1 = |D| = γL(G).
This and the following lemma provide an alternative algorithm for the k-domination problem in trees.
Lemma 2. Suppose G is a graph with a k-dom assignment L = (L1, L2).
For a leaf v of G adjacent to u, let G′ = G− v and let L′ be the restriction of L on V (G′) with the modification described below.
(1) If L(v) = (B, 1), then γL(G) = γL′(G′), where L′1(u) = R.
(2) If L(v) = (B, 0), then γL(G) = γL′(G′).
Proof. (1) Suppose D′ is a γL′-set of G′. Since L′1(u) = R, we have u∈ D′. Then D′ is an L-dominating set of G as |N(v) ∩ D′| ≥ 1 = L2(v). Thus γL(G)≤ |D′| = γL′(G′).
Conversely, suppose D is a γL-set of G. Since L2(v) = 1, either u or v must be included in D. Then clearly D′ = (D\ { v }) ∪ { u } is an L′-dominating set of G′. Hence γL′(G′)≤ |D′| ≤ |D| = γL(G).
(2) Suppose D′ is a γL′-set of G′. Since L′ is the restriction of L on G′ and |N(v) ∩ D′| ≥ 0 = L2(v), it is clear that D′ is an L-dominating set of G.
Thus γL(G)≤ |D′| = γL′(G′).
Conversely, suppose D is a γL-set of G. If v ̸∈ D, then D′ = D is an L′-dominating set of G′. If v ∈ D, then D′ = (D\ { v }) ∪ { u } is an L′-dominating set of G′. Hence γL′(G′)≤ |D′| ≤ |D| = γL(G).
Having these two lemmas at hand, we now establish an alternative al- gorithm for L-domination in trees. The algorithm will also be used later as a subroutine to find a minimum L-dominating set for cycles, cacti and block-cactus graphs.
Given a tree T of n vertices, it is well-known that T has a vertex ordering v1, v2, . . . , vnsuch that vi is a leaf of Gi = G[vi, vi+1, . . . , vn] for 1≤ i ≤ n−1.
This ordering can be found in linear-time by using, for example, the bread- first-search (BFS) algorithm.
The algorithm starts processing a leaf v of a tree T , which is adjacent to a unique vertex u. The label of v is used to possibly relabel u. After v is visited, v is removed from T and obtain a new tree T′. A linear-time labeling algorithm for finding a γL-set in trees is shown as follows.
Algorithm: kDomTree (Finding a γL-set of a tree)
Input: A tree T of n vertices with a tree ordering v1, v2, . . . , vn and a k-dom assignment L = (L1, L2).
Output: A minimum L-dominating set D of T . Method:
D← ∅;
for i ← 1 to n do
let u be the parent of vi (regard u as vi if i = n);
if L1(vi) = R or L2(vi) > 1 then L2(u)← max { L2(u)− 1, 0 };
D← D ∪ { vi};
end
if L(vi) = (B, 1) then L1(u)← R;
end end
Theorem 3. Algorithm kDomTree finds a minimum L-dominating set for a tree in linear-time.
4. k-Domination for graphs with special blocks
The main result of this section is an algorithm for the k-domination prob- lem in graphs with tree-like structure. More precisely, we present a linear- time algorithm to find a minimum k-dominating set of a graph whose blocks are cliques, cycles or complete bipartite graphs. These include block graphs, cacti and block-cactus graphs, etc.
Suppose G is a graph with a k-dom assignment L = (L1, L2). For a vertex v of G, denote by D∗v(G) a minimum L-dominating set of G such that v has the most neighbors in this set, i.e., |N(v) ∩ Dv∗(G)| is maximum. Fig. 1(a) illustrates an example of Dv∗(G), while the minimum L-dominating set forms by the shaded vertices in Fig. 1(b) cannot be selected as Dv∗(G).
(B,0) (B,3) (B,2)
(B,2) v
(B,0) (B,3) (B,2)
(B,2) v
(a) (b)
Figure 1: Minimum L-dominating sets in a graph.
Let C be an end-block of G and x be its unique cut-vertex. Denote the end-block C with the modification on L1(x) = R by CR. Denote the end- block C with the modification on L2(x) = 0 by C0. Note that γL′(C0) ≤ γL′(C) ≤ γL′(CR) ≤ γL′(C0) + 1, where L′ is the restriction of L on V (C) with those modifications.
The construction and correctness of the algorithm is based on the follow- ing theorem.
Theorem 4. Suppose G is a graph with a k-dom assignment L = (L1, L2).
Let C be an end-block of G and let x be the unique cut-vertex of C. Let G′ denote the graph which results from G by deleting all vertices only in C. Let L′ and L′′ be the restriction of L on G′ and C with modifications as described below.
(1) If L1(x) = R or L2(x) > degG(x), then γL(G) = γL′(G′) + γL′′(C)− 1.
(2) If L1(x) = B and γL′′(C0) < γL′′(CR), then γL(G) = γL′(G′) + γL′′(C), where L′2(x) = max{L2(x)− t, 0}, L′′2(x) = t and t =|N(x) ∩ D∗x(C0)|.
(3) If L1(x) = B and γL′′(C0) = γL′′(CR), then γL(G) = γL′(G′)+γL′′(C)−1, where L′1(x) = L′′1(x) = R.
Proof. (1) Let D1be a γL′-set of G′ and D2be a γL′′-set of C. Since L′1(x) = R or L′2(x) > degG′(x), x ∈ D1; and since L′′1(x) = R or L′′2(x) > degC(x), x∈ D2. Clearly D1∪ D2 is an L-dominating set of G and we have γL(G)≤
|D1∪ D2| = γL′(G′) + γL′′(C)− 1.
Conversely, suppose D is a γL-set of G. By the assumption that L1(x) = R or L2(x) > degG(x), we have x ∈ D. It is clear that D ∩ V (G′) is an L′- dominating set of G′ and D ∩ V (C) is an L′′-dominating set of C. Hence γL′(G′) + γL′′(C)− 1 ≤ |D ∩ V (G′)| + |D ∩ V (C)| − 1 = |D| = γL(G).
(2) Let D1be a γL′-set of G′and D2 be a γL′′-set of C. If x ∈ D1∪D2, then clearly D1∪ D2 is an L-dominating set of G and hence γL(G)≤ |D1 ∪ D2| ≤ γL′(G′) + γL′′(C). Suppose x̸∈ D1∪ D2. Then
|N(x) ∩ (D1∪ D2)| = |N(x) ∩ D1| + |N(x) ∩ D2|
≥ L′2(x) + L′′2(x)
≥ L2(x)− t + t = L2(x).
Thus D1∪D2is an L-dominating set of G and hence γL(G)≤ γL′(G′)+γL′′(C).
Conversely, suppose D is a γL-set of G. We have two cases.
Case 1: x ∈ D. By the assumption that γL′′(C0) < γL′′(CR), we have
|D ∩ V (C)| > γL′′(C0). Let D′ = (D− V (C)) ∪ Dx∗(C0)∪ { x }. Then |D′| =
|D| and D′ is also an L-dominating set of G. Clearly (D− V (C)) ∪ { x } is an L′-dominating set of G′. Since |N(x) ∩ D∗x(C0)| = t ≥ L′′2(x), Dx∗(C0) is an L′′-dominating set of C. Thus γL′(G′) + γL′′(C)≤ |(D − V (C)) ∪ { x }| +
|D∗x(C0)| = |D′| = |D| = γL(G).
Case 2: x ̸∈ D. Then D ∩ V (C) must contain an L′′-dominating set of C0. Thus |D ∩ V (C)| ≥ γL′′(C0). Since D is a γL-set of G, we have
|N(x) ∩ D| ≥ L2(x). Let D′ = (D − V (C)) ∪ Dx∗(C0). Then |D′| = |D|
and D′ is also an L-dominating set of G. Since |N(x) ∩ (D − V (C))| ≥ L2(x) − |D ∩ V (C)| ≥ L2(x)− t = L′2(x), it is clear that D − V (C) is an L′-dominating set of G′; since |N(x) ∩ Dx∗(C0)| = t ≥ L′′2(x), it is also that D∗x(C0) is an L′′-dominating set of C. Therefore γL′(G′) + γL′′(C) ≤
|D − V (C)| + |Dx∗(C0)| = |D′| = |D| = γL(G).
(3) Let L be the same as L except for the modification on L1(x) = R. We claim that under the assumptions of this case, γL(G) = γL(G). If the claim is true, then by (1), we have the desire result. By definition, clearly γL(G)≥ γL(G). Let D be a γL-set of G. By the assumption that γL′′(C0) = γL′′(CR), one can always replace the elements in D∩ V (C) by a γL′′-set of CR to get an L-dominating set of G. Thus γL(G) ≤ |D| = γL(G).
We are now ready to present our algorithm, called kDomG, to determine a minimum L-dominating set of a graph. Our algorithm takes kDomB as a subroutine, which we assume it can find a minimum L′-dominating set of each end-block C of a graph, where L′ is the restriction of L on V (C).
Algorithm: kDomG (A general approach for finding a γL-set in graphs and kDomB is a subroutine we assume that can find a γL′-set of each end-block of the graph)
Input: A graph G with a k-dom assignment L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
G′ ← G;
D← ∅;
while G′ ̸= ∅ do
if G′ is a block then D← D ∪ kDomB(G′);
G′ ← ∅;
else
let C be an end-block of G′ and x be its unique cut-vertex;
if L1(x) = R or L2(x) > deg(x) then D← D ∪ kDomB(C);
else
U0 ← kDomB(C0);
UR ← kDomB(CR);
if |U0| < |UR| then D← D ∪ Dx∗(C0);
L2(x)← max{L2(x)− |N(x) ∩ D∗x(C0)|, 0};
else // |U0| = |UR| D← D ∪ UR; L1(x)← R;
G′ ← G′− (V (C) − { x });
end
Theorem 5. Algorithm kDomG finds a minimum L-dominating set of a graph G in linear-time if kDomB and D∗x(C0) takes linear time to compute for each end-block C of G with cut-vertex x.
Proof. The correctness comes from Theorem 4. For the time complexity, since kDomG calls at most three times of kDomB and computes D∗0at most once for each end-block of the graph, it is clear that kDomG is linear.
5. The implementations of the subroutine kDomB in graphs In this section, we show how to implement the subroutine kDomB for some classes of graphs. In particular, we present linear-time algorithms for finding a minimum L-dominating set for complete graphs, cycles and com- plete bipartite graphs. In addition, the computations of Dx∗(G) for the men- tioned classes of graphs are also discussed.
Throughout the rest of this section, suppose G = (V, E) is a graph with a k-dom assignment L = (L1, L2). Define
R =e { v ∈ V : L1(v) = R or L2(v) > degG(v)} and let | eR| = r.
Let D be a minimum L-dominating set of G. By the definition of L- dominating set, all vertices of eR must be included in D.
5.1. Complete graphs
Assume G is a complete graph. Suppose |V | = n and | eR| = r. If ui ̸∈ D and uj ∈ D with L2(ui) > L2(uj) for some ui, uj ∈ V \ eR, then (D−uj)∪uiis also a minimum L-dominating set of G. This indicates that we shall choose vertices in V \ eR with L2 value as large as possible. Let t be the minimum index in { 0, 1, . . . , n − r } such that t ≥ L2(ut+1)− r. It is the case that D = eR∪ { ui ∈ V \ eR : 1 ≤ i ≤ t }.
Algorithm: kDomKn (Finding a γL-set of a complete graph) Input: A complete graph G = (V, E) with a k-dom assignment
L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
let u1, u2, . . . , un−r be a vertex ordering of V \ eR such that L2(u1)≥ L2(u2)≥ . . . ≥ L2(un−r);
u0 ← ∅;
L2(un−r+1)← r;
for t = 0 to n− r do
if t≥ L2(ut+1)− r then break;
end
D← eR∪ { ui ∈ V \ eR : 0≤ i ≤ t };
Theorem 6. Algorithm kDomKn finds a minimum L-dominating set for a complete graph in linear time.
Proof. The correctness is clear and is omitted. The time complexity is bound by the computation of the vertex ordering of V \ eR. Note that L2(v) ≤ degG(v) < n for all v ∈ V \ eR. Since each L2(v) is an integer in the range 0 to n and there are at most n integers need to sort, one can use linear-time sorting algorithms, for examples, Counting sort or Bucket sort, to obtain the vertex ordering.
Consider the computation of D∗x(G) of G for some fixed vertex x. Since each pair of vertices of G is adjacent, any minimum L-dominating set D of G has the property that|N(x) ∩ D| is maximum, and can be selected as D∗x(G).
Thus Dx∗(G) can be found in linear time.
5.2. Cycles
Assume G is a cycle. We will use kDomTree, introduced in Section 3, as a subroutine to find a minimum L-dominating set of G. First consider the case of eR ̸= ∅. Let v be a vertex in eR. Since v must be included in D, the L2 values of the neighbors of v should be decreased by 1. Let Pv be the path of C − v with the mentioned modifications on the L2 values of the neighbors of v. By Lemma 1, any minimum L′-dominating set of Pv and v form a minimum L-dominating set of G.
Now assume eR = ∅. If L2(v) = 0 for all v ∈ V , then clearly D = ∅.
Otherwise suppose L2(v) > 0 for some v ∈ V . Let u, w be the two neighbors of v on C. In this case, either v is in D or at least one of u and w is in D. Thus,
|D| = min{|kDomTree(Pv)| , |kDomTree(Pu)| , |kDomTree(Pw)|} + 1.
The time complexity is clearly linear, since kDomTree is linear and it calls at most three times of kDomTree.
Now consider the computation of Dx∗(G) of G for some fixed vertex x. It is obvious that|N(x) ∩ Dx∗(G)| ≤ 2. Thus one can find D∗x(G) by examining among minimum L-dominating sets of G with all possible combinations of modifications on L1(y) = R, where y ∈ N[x]. The computation of D∗x(G) clearly can be done in linear time.
Algorithm: kDomCYC (Finding a γL-set set of a cycle)
Input: A cycle G = (V, E) with a k-dom assignment L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
if eR̸= ∅ then choose v∈ eR;
let Pv be the path of G− v with the modifications on L2(u)← max{L2(u)− 1, 0} for each u ∈ N(v);
D← kDomTree(Pv)∪ { v };
else
if there is a vertex v such that L2(v) > 0 then foreach u∈ N[v] do
Du ← kDomCYC(GRu), where GRu is the same as G with the modification on L1(u) = R;
end
D← Du with minimum |Du|, u ∈ N[v];
end
Theorem 7. Algorithm kDomCYC finds a minimum L-dominating set in a cycle in linear time.
5.3. Complete bipartite graphs
Now consider that G is a complete bipartite graph whose vertex set is a disjoint union of two independent sets A and B. If | eR∩ A| is no less than L2(b) for all b ∈ B \ eR and | eR∩ B| is no less than L2(a) for all a ∈ A \ eR, then each vertex v not in eR has at least L2(v) neighbors in eR. Otherwise, D must contain some vertices in (A∪ B) \ eR. Again, if D contains some ai (resp. bi), then it is better to choose ai (resp. bi) with L2(ai) (resp. L2(bi)) as large as possible. Suppose D contains exactly i vertices in A\ eR, where 0 ≤ i ≤ |A| − r1. Then D must contains at least j vertices in B \ eR and L2(bj+1) − r1 ≤ i, where j = L2(ai+1)− r2. In addition, we can assume D∩ (A \ eR) = { a0, . . . , ai}, where a0 = ∅. Since the algorithm examines all possible choices of i, D is clearly a minimum L-dominating set of G. See Fig. 2 for illustrations.
a1 a
2 ai ai
+1
ܣځܴ෩
ܤځܴ෩ r1
r2
L2(a1)-r2 L
2(a2)-r2 ܮଶܽ െݎଶ ܮଶܽାଵ െݎଶ
a|A|-r1
ڮ ڮ
ڮ
݆ = ܮଶܽାଵ െݎଶ ܮଶܾାଵ െݎଵ
ܣ
ܤ
Figure 2: Finding a minimum L-dominating set in a complete bipartite graph.
Algorithm kDomKmn (Finding a γL-set of a complete bipartite graph) Input: A complete bipartite graph G whose vertex set is a disjoint union of
two independent sets A and B, and a k-dom assignment L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
if r1 ≥ max{L2(b) : b ∈ B \ eR} and r2 ≥ max{L2(a) : a∈ A \ eR} then D← eR;
else
r1 ← |A ∩ eR|; r2 ← |B ∩ eR|;
let a1, a2, . . . , a|A|−r1 and b1, b2, . . . , b|B|−r2 be vertex orderings of A\ eR and B\ eR, respectively, such that L2(a1)≥ L2(a2)≥ . . . ≥ L2(a|A|−r1) and L2(b1)≥ L2(b2)≥ . . . ≥ L2(b|B|−r2);
a0 ← ∅; b0 ← ∅; L2(a|A|−r1+1)← r2; L2(b|B|−r2+1)← r1; size← ∞;
for i = 0 to |A| − r1 do j ← L2(ai+1)− r2;
if i + j < size and L2(bj+1)− r1 ≤ i then t ← i;
size ← i + j;
end end
D← eR∪ { ai ∈ A \ eR : 0≤ i ≤ t } ∪ { bi ∈ B \ eR : 0≤ i ≤ L2(at+1)− r2};
Theorem 8. Algorithm kDomKmn finds a minimum L-dominating set in a complete bipartite graph in linear time.
Proof. The correctness is clear and is omitted. The time complexity is linear, since the vertex orderings of A\ eR and B\ eR can be found by using linear- time sorting algorithms, and it takes O(|V |) time on the for-loop of the algorithm.
Now consider how to compute Dx∗(G) of G for some fixed vertex x.
W.L.O.G., suppose x ∈ A. First run kDomKmn to get the size, say d, of a γL-set of G. Then D∗x(G) can be found by checking the validity of Re∪ { a0, . . . , ai} ∪ { b0, . . . , bd−r−i} for all 0 ≤ i ≤ d − r, and picking the set with maximum d− r − i. The process clearly can be done in linear time.
It is well-known that block graphs, cacti, block-cactus graphs can be recognized in linear-time. By Theorems 5, 6, 7, and 8, one can immediately have the following result.
Theorem 9. Algorithm kDomG finds a minimum L-dominating set in lin- ear time for graphs in which each block is a clique, a cycle or a complete bipartite graph, including block graphs, cacti and block-cactus graphs.
Fig. 3 shows an example of k-domination in a graph in which the graph contains complete graphs, cycles, and complete bipartite graphs as blocks.
Figure 3: k-domination in a graph with special blocks; k = 3.
6. NP-completeness results
In this section, we study the complexity of the k-domination problem:
k-DOMINATION
INSTANCE: A graph G = (V, E) and positive integers k and s.
QUESTION: Does G have a k-dominating set of size ≤ s?
It has been proved that the k-domination is NP-complete for general graphs [14], bipartite graphs [1] and chordal graphs [1], in which the reduc- tions are mainly from domination problem for the same class of graphs. In this section, we show that the k-domination remains NP-complete for split graphs, a subclass of chordal graphs. A split graph is a graph whose vertex set is the disjoint union of a clique and a stable set. Our reduction is from a well-known NP-complete problem, vertex cover problem for general graphs.
A vertex cover of a graph G = (V, E) is a subset C ⊆ V such that for every edge uv ∈ E we have u ∈ C or v ∈ C. The vertex cover problem is to find a minimum vertex cover of G.
VERTEX-COVER
INSTANCE: A graph G = (V, E) and a nonnegative integer c.
QUESTION: Does G have a vertex cover of size ≤ c?
Theorem 10. For any fixed positive integer k, k-DOMINATION is NP- complete for split graphs.
Proof. Obviously k-DOMINATION belongs to NP, since it is easy to verify a “yes” instance of k-DOMINATION in polynomial time. The reduction is from the vertex cover problem. Let G = (V, E) be an instance of VERTEX- COVER. We construct the graph H = (VH, EH) with vertex set VH = V ∪ VE ∪ U ∪ W ∪ { x }, where VE = { ve: e∈ E }, U = { u1, u2, . . . , uk−1}, W ={ w1, w2, . . . , uk}, and edge set
EH = { vve: v ∈ V, ve∈ VE, v ∈ e }
∪ { uive: ui ∈ U, ve ∈ VE}
∪ { uv : u ∈ U ∪ { x } , v ∈ W }
∪ { uv : u, v ∈ V ∪ U ∪ { x } , u ̸= v } .
Clearly, H is a split graph whose vertex set is a disjoint union of a clique V∪U ∪{ x } and a stable set VE∪W , and H can be constructed in polynomial time. Now we shall show that G has a vertex cover of size c if and only if H has a k-dominating set of size c + k.
Suppose G has a vertex cover C of size c. Then choose DH = C∪U ∪{ x }.
It is not hard to verify that DH is a k-dominating set of H of size c + k.
On the other hand, suppose DH is a k-dominating set of H of size c + k.
If wj ̸∈ DH for some wj ∈ W , then DH must contain x and all vertices of U . If DH contains all vertices of W , then (DH \ W ) ∪ U ∪ { x } is also a k-dominating set. Thus, we may assume DH contains x and all vertices of U . Set D = DH∩ (V ∪ VE). We have|D| ≤ c. If D contains some ve ∈ VE, then (D− ve)∪ { v } is also a k-dominating set of H for some v ∈ V, e ∈ E, v ∈ e.
Thus, we may assume D∩ VE =∅. As a result, adding enough vertices to D results in a vertex cover of G size c.
v1 v2 v3 v4 u1 u2 x
G H
e1 e2 e3 v1 v2 v3 v4
w2 w1
w3 ve
2
ve
1
ve
3
Figure 4: A transformation to a split graph when k = 3.
References
[1] T.J. Bean, M.A. Henning, H.C. Swart, On the integrity of distance domination in graphs, Australas. J. Combin. 10 (1994) 29–43.
[2] Y. Caro, Y. Roditt, A note on the k-domination number of a graph, Int.
J. Math.Math. Sci. 13 (1990) 205–206.
[3] G.J. Chang, Algorithmic aspects of domination in graphs, in: in: Hand- book of Combinatorial Optimization (D.-Z. Du and P. M. Pardalos eds, pp. 339–405.
[4] M. Chellali, O. Favaron, A. Hansberg, L. Volkmann, On the p- domination, the total domination and the connected domination num- bers of graphs, J. Combin. Math. Combin. Comput. 73 (2010) 65–75.
[5] M. Chellali, O. Favaron, A. Hansberg, L. Volkmann, k-Domination and k-independence in graphs: a survey, Graphs and Combinatorics 28 (2012) 1–55.
[6] B. Chen, S. Zhou, Upper bounds for f -domination number of graphs, Discrete Math. 185 (1998) 239–243.
[7] E.J. Cockayne, B. Gamble, B. Shepherd, An upper bound for the k- domination number of a graph, J. Graph Theory 9 (1985) 533–534.
[8] E.J. Cockayne, S.E. Goodman, S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inf. Process. Lett. 4 (1975) 41–44.
[9] O. Favaron, A. Hansberg, L. Volkmann, On k-domination and minimum degree in graphs, J. Graph Theory 57 (2008) 33–40.
[10] J.F. Fink, M.S. Jacobson, Graph theory with applications to algorithms and computer science, John Wiley & Sons, Inc., New York, NY, USA, 1985, pp. 283–300.
[11] A. Hansberg, L. Volkmann, Upper bounds on the k-domination number and the k-Roman domination number, Discrete Appl. Math. 157 (2009) 1634–1639.
[12] T. Haynes, S. Hedetniemi, P. Slater, Fundamentals of domination in graphs, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, 1998.
[13] S. Hedetniemi, R. Laskar, J. Pfaff, A linear algorithm for finding a minimum dominating set in a cactus, Discrete Applied Mathematics 13 (1986) 287–292.
[14] M.S. Jacobson, K. Peters, Complexity questions for n-domination and related parameters, Congr. Numer. 68 (1989) 7–22. Eighteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1988).
[15] C.S. Liao, G.J. Chang, Algorithmic aspect of k-tuple domination in graphs, Taiwanese J. Math 6 (2003) 415–420.
[16] C. Stracke, L. Volkmann, A new domination conception, J. Graph The- ory 17 (1993) 315–323.
