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國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University
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On the algorithmic complexity of k-tuple total domination
James K. Lan and Gerard Jennhwa Chang
May 29, 2013
On the algorithmic complexity of k-tuple total domination
IJames K. Lana,∗, Gerard Jennhwa Changa,b,c
aDepartment of Mathematics, National Taiwan University, Taipei 10617, Taiwan
bTaida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan
cNational Center for Theoretical Sciences, Taipei Office, Taiwan
Abstract
For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G is adjacent to at least k vertices in D. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of G. This paper studies k-tuple total domination from an algorithmic point of view. In particular, we present a linear-time algorithm for the k-tuple total domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph, which include trees, block graphs, cacti and block-cactus graphs. We also establish NP-completeness of the k-tuple total domination problem in undirected path graphs.
Keywords: k-Tuple total domination, Total domination, Block graph, Cactus, Algorithm, NP-complete, Undirected path graph
1. Introduction
All graphs in this paper are simple, i.e., finite, undirected, loopless and without multiple edges. Domination is a well studied subject in graph theory and combinatorial optimization as it has many applications in the real world such as location problems, sets of representatives, social network theory, etc
IThis research was partially supported by the National Science Council of the Republic of China under grants NSC100-2811-M-002-146 and NSC101-2115-M-002-005-MY3.
∗Corresponding author.
Email addresses: drjamesblue@gmail.com (James K. Lan), gjchang@math.ntu.edu.tw (Gerard Jennhwa Chang)
and the literature on this topic has been surveyed and detailed in books [2, 5, 6]. A dominating set of a graph G is a subset D ⊆ V (G) such that every vertex not in D has at least one neighbor in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination problem is to find a minimum dominating set of a graph.
The idea of dominating all vertices of the graph, rather than merely dominating vertices outside the set, is considered by Cockayne, Dawes and Hedetniemi [3]. A total dominating set of a graph G is a subset D ⊆ V (G) such that every vertex of G has at least one neighbor in D. Every graph without isolated vertices has a total dominating set, since the vertex set V (G) is such a set. The total domination number γt(G) of G is the minimum cardinality of a total dominating set of G. Total domination is now a well- studied topic in graph theory; see the recent survey paper [7] for more details.
Another variation of domination, the k-tuple domination was introduced by Harary and Haynes [4]. For a fixed positive integer k, a k-tuple dominating set of a graph G is a subset D ⊆ V (G) such that the closed neighborhood of every vertex of G has at least k vertices in D. The k-tuple domination number γ×k(G) of G is the minimum cardinality of a k-tuple dominating set of G. The k-tuple domination number is only defined for graphs with minimum degree at least k − 1. The special case when k = 1 is the usual domination.
Motivated by the concept of k-tuple domination, Henning and Kazemi [8]
considered the following generalization of total domination. For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset D⊆ V (G) such that every vertex of G has at least k neighbors in D. Every graph with minimum degree at least k admits a k-tuple total dominating set.
The k-tuple total domination number γ×k,t(G) of G is the minimum cardi- nality of a k-tuple total dominating set of G. The k-tuple total domination problem is to determine a minimum k-tuple total dominating set of a graph.
Since a (k + 1)-tuple dominating set is also a k-tuple total dominating set and a k-tuple total dominating set is also a k-tuple dominating set, we have γ×k(G) ≤ γ×k,t(G) ≤ γ×(k+1)(G) for graphs with minimum degree at least k. The 1-tuple total domination is the well-studied total domination. The 2-tuple total domination is called the double total domination in the litera- ture. Authors in [8, 9, 10] have established bounds on the number γ×k,t(G) in terms of different graph invariants.
On the complexity side of the k-tuple total domination problem, Prad- han [14] showed that the k-tuple total domination problem is NP-complete
for bipartite graphs and for split graphs (and thus the chordal graphs). This problem remains NP-complete for doubly chordal graphs, another subclass of chordal graphs [14]. Apart from these, Pradhan also proposed some hardness results and approximation algorithms for this problem. On the other hand, Pradhan proved that the k-tuple total domination problem for chordal bipar- tite graphs is a subproblem of the k-tuple domination problem for strongly chordal graphs, which is solvable in polynomial time [13]. Therefore the k- tuple total domination problem for chordal bipartite graphs is polynomially solvable.
In this paper, we explore efficient algorithms for the k-tuple total domi- nation problem in graphs. In particular, we present a linear-time algorithm for the k-tuple total 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. Since the k-tuple total domination is a generalization of the total domination, our algorithms cover the partial results in [1, 7, 12]. Moreover, we also show that the k-tuple total domination problem remains NP-complete for undirected path graphs, another 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 NG(v) = { u ∈ V : uv ∈ E } and the closed neighborhood is NG[v] = NG(v)∪ { v }. The degree degG(v) of a vertex v in G is the number of edges incident to v. When the graph G is clear from the context, it is dropped from notations. The minimum degree among the vertices of G is denoted by δ(G). An isolated vertex is a vertex v with deg(v) = 0. A leaf of a graph is a vertex with degree one. 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, an independent set is a set of pairwise nonadjacent vertices; a clique is a set of pairwise adjacent vertices. A tree is a connected graph with- out cycles. 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 con- taining at most one cut-vertex of the graph. 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 cactus is a tree if all the blocks are edges. A block-cactus graph is a graph whose blocks are cliques or cycles.
A graph is an intersection graph if there is a correspondence between its vertices and a family of sets (the intersection model) such that two distinct vertices are adjacent in the graph if and only if their two corresponding sets have a nonempty intersection. A graph G is chordal if G has no induced cycle with length greater than 3. It is well-known that a graph is chordal if and only if it is the intersection graph of some subtrees of a certain tree.
If these subtrees are paths, this chordal graph is called an undirected path graph.
3. Block-wise approach for k-tuple total domination
The main result of this section is an algorithm for the k-tuple total dom- ination problem in graphs. Actually our algorithm solves a slightly more general problem, which will be formulated as “L-domination”.
3.1. Labeling method for k-tuple total domination
Labeling techniques are widely used in the literatures for solving the domi- nation problem and its variants [2, 11, 12, 13]. For k-tuple total domination, we employ the following labeling method which is similar to that in [11].
Given a graph G, labeling L is a mapping 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
• for each v ∈ V (G), |NG(v)∩ D| ≥ L2(v).
That is, D contains all required vertices, and for each vertex v of G, v is adjacent to at least L2(v) vertices in D. The L-domination number γL(G) is the minimum cardinality of an L-dominating set in G, such set is called a γL-set of G. Clearly G has an L-dominating set if and only if L2(v)≤ degG(v) for all v ∈ V (G), and we call such L a proper labeling. Notice that if L(v) = (B, k) for all v ∈ V (G), then γL(G) = γ×k,t(G). Thus an algorithm for γL(G)
gives γ×k,t(G). In the following, we give a general approach (block-wise) to find a minimum L-dominating set in a graph.
Suppose G is a graph with a proper labeling L = (L1, L2). Let C be an end-block of G and x be its unique cut-vertex. Let G′ denote the graph which results from G by deleting all vertices only in C. Suppose ℓ is a nonnegative integer such that ℓ ≤ degC(x). The following notations will be used throughout the rest of this section.
• Lℓ: the restriction of L on C with the modification on Lℓ2(x) = ℓ.
• LℓR: the same as Lℓ except for the modification on LℓR1 (x) = R.
Since the value L2(x) may be greater than the number of neighbors of x in C, we need to set the L2 value of x as the minimum between L2(x) and degC(x) before evaluating the cardinality of a γL-set of C. Thus, for convenience, set
α = min{L2(x), degC(x)}.
Note that one always have γL0(C) ≤ γLℓ(C) ≤ γLℓR(C). Especially, substi- tuting α for ℓ gives
γL0(C)≤ γLα(C)≤ γLαR(C). (1) For a vertex v of G with a proper labeling L, denote by D∗v(G, L) a min- imum L-dominating set of G such that v has the most neighbors in this set among all minimum L-dominating sets of G, i.e., |NG(v)∩ Dv∗(G, L)| ≥
|NG(v)∩ D| for any minimum L-dominating set D of G. Fig. 1 (b) illustrates an example of D∗v(G, L), while the minimum L-dominating set formed by the shaded vertices in Fig. 1 (a) cannot be selected as a Dv∗(G, L). The construc- tion and correctness of the algorithm is based on the following theorem.
(B,0) (B,1) (B,1)
v
(a) (b)
(B,2) (B,0)
(B,0) (B,1) (B,1)
v
(B,2) (B,0)
Figure 1: Minimum L-dominating sets of a graph.
Theorem 1. Suppose G is a graph with a proper labeling L = (L1, L2).
Suppose C is an end-block of G and x is its unique cut-vertex. Let G′ = (G− C) ∪ { x } and L′ be the restriction of L on G′ with modifications as described below. Let α = min{L2(x), degC(x)}, s = |NC(x)∩ D∗x(C, L0)| and t = max{L2(x)− s − degG′(x), 0}.
(1) If γL0(C) = γLαR(C), then γL(G) = γL′(G′)+γLαR(C)−1, where L′1(x) = R, L′2(x) = L2(x)− α.
(2) If γL0(C) = γLα(C) < γLαR(C), then γL(G) = γL′(G′) + γLα(C), where L′2(x) = L2(x)− α.
(3) Suppose γL0(C) < γLα(C).
(3.1) If L1(x) = R or there exists y ∈ NG(x) such that L2(y) = degG(y), then γL(G) = γL′(G′) + γLℓR(C)− 1, where L′1(x) = R, L′2(x) = L2(x)− ℓ and ℓ = s + t.
(3.2) if L2(y) < degG(y) for all y ∈ NG(x), then γL(G) = γL′(G′) + γLℓ(C), where L′2(x) = L2(x)− ℓ and ℓ = s + t.
The following observation is easily obtained.
Proposition 2. Suppose G is a graph with a proper labeling L. Let D be a γL-set of G. Then |D ∩ C| ≥ γL0(C).
Lemma 3. Suppose G is a graph with a proper labeling L. Let C be an end-block of G, x be its unique cut-vertex and G′ = (G− C) ∪ { x }. Let D be a γL-set of G. Let α = min{L2(x), degC(x)}. Let DC be a γLα-set or a γLαR-set of C and D = (D− C) ∪ DC. Then NG′(x)∩ D ≥L2(x)− α and NC(x)∩ D ≥α. In other words, NG(x)∩ D ≥L2(x).
Proof. By the definition of L-dominating set, |NG(x)∩ D| ≥ L2(x) and NC(x)∩ D ≥α. If L2(x)≤ degC(x), i.e., α = L2(x), then NG′(x)∩ D ≥ 0 = L2(x)− α. If L2(x) > degC(x), i.e., α = degC(x), then α = degC(x)≥
|NC(x)∩ D|, as x can have at most degC(x) neighbors in D. Hence|NG′(x)∩ D| = |NG′(x)∩ D| = |NG(x)∩ D| − |NC(x)∩ D| ≥ L2(x)− α. The assertion holds clearly.
Proof of Theorem 1 (1). Let D′ be a γL′-set of G′ and DC be a γLαR-set of C. Since L′1(x) = LαR1 (x) = R, x∈ D′∪DC. Also that|NG(x)∩ (D′∪ DC)| =
|NG(x)∩ D′| + |NG(x)∩ DC| ≥ L′2(x) + α = L2(x), by the assumption. We have D′ ∪ DC is an L-dominating set of G and hence γL(G) ≤ |D′ ∪ DC| =
|D′| + |DC| − 1 = γL′(G′) + γLαR(C)− 1.
Conversely, suppose D is a γL-set of G. Let DC be a γLαR-set of C and let D = (D− C) ∪ DC. By the assumption that γL0(C) = γLαR(C) and Eq. (1) and Proposition 2,|D| ≥ D . Note that no matter what L1(x) might be, x is always be included in D. By Lemma 3, NG(x)∩ D ≥L2(x) and thus D is also an L-dominating set of G. Moreover, NG′(x)∩ D ≥L2(x)−α = L′2(x) and NC(x)∩ D ≥α. Thus D∩ G′ is an L′-dominating set of G′ and D∩ C is an LαR-dominating set of C. Hence γL(G) = |D| ≥ D = D∩ G′ + D∩ C −1≥ γL′(G′) + γLαR(C)− 1.
Note that under the assumptions of Theorem 1 (2), x must be a bound vertex, i.e., L1(x) = B. The argument is similar to that of Theorem 1 (1) and therefore is omitted.
Proposition 4. Suppose G is a graph that is also a block. Suppose L is a proper labeling on G. Let x be a vertex of G. Let s = |NG(x)∩ D∗x(G, L0)| and j be a nonnegative integer such that s + j ≤ degG(x). Then γLs+j(G) = γLs(G) + j.
Proof. Let Dsbe a γLs-set of G and D = Ds∪J, where J consists of j vertices of NG(x)\ Ds. Clearly D is an Ls+j-dominating set of G and γLs+j(G) ≤
|D| = |Ds| + j = γLs(G) + j holds.
Conversely, suppose D is a γLs+j-set of G and we have |NG(x)∩ D| ≥ s + j. Since D must contain an L0-dominating set of G, there exists a subset J ⊆ V (G) such that D \ J is an L0-dominating set of G. Clearly,
|J| ≤ γLs+j(G)− γL0(G). Also, by the definition of s, γL0(G) = γLs(G) and
|NG(x)∩ (D \ J)| ≤ s. This indicates that |J| ≥ j. Therefore, γLs+j(G) ≥ γL0(G) + j = γLs(G) + j.
Proof of Theorem 1 (3). We first consider (3.1) and assume that L1(x) = R. It is easy to check that LℓR is a proper labeling on V (C). Let D′ be a γL′-set of G′ and DC be a γLℓR-set of C. Since L′1(x) = LℓR1 (x) = R, x∈ D′∪ DC. Also by assumptions, we have |NG(x)∩ (D′∪ DC)| = |NG(x)∩ D′| +
|NG(x)∩ DC| ≥ L2(x). Thus D′∪DC is an L-dominating set of G and hence γL(G)≤ |D′∪ DC| = |D′| + |DC| − 1 = γL′(G′) + γLℓR(C)− 1.
Conversely, suppose D is a γL-set of G. Let DC be a γLℓR-set of C.
Case 1: |NC(x)∩ D| ≤ ℓ. Let D = (D − C) ∪ DC. We now claim that
|D ∩ C| ≥ γLℓR(C). If this claim holds, then we have |D| ≥ D . First consider the case of t = 0. Since ℓ = s + t, we have γLℓR(C) = γLsR(C).
Since x can have at most s neighbors in a γL0R-set of C, γLsR(C) = γL0R(C).
By Proposition 2 and the assumption that L1(x) = R, |D ∩ C| ≥ γL0(C) = γL0R(C) = γLℓR(C) holds.
Now suppose t > 0. From the definition of t, we have L2(x) = ℓ+degG′(x).
Since D contains at least L2(x) vertices of NG(x), D must contain at least ℓ vertices of NC(x), i.e.,|NC(x)∩ D| ≥ ℓ, for otherwise we have |NG′(x)∩ D| >
L2(x)− ℓ = degG′(x), a contradiction. By the assumption that L1(x) = R, D∩ C is also an LℓR-dominating set of C. Hence |D ∩ C| ≥ γLℓR(C). We therefore have this claim.
By the choice of D, x is included in D. Also that NC(x)∩ D ≥ ℓ and thus D ∩ C is an LℓR-dominating set of C. By the assumption that
|NC(x)∩ D| ≤ ℓ, we have NG′(x)∩ D = |NG(x)∩ D| − |NC(x)∩ D| ≥ L2(x)− ℓ = L′2(x), and thus D∩ G′ is also an L′-dominating set of G′. Hence γL(G) =|D| ≥ D = D∩ G′ + D∩ C −1≥ γL′(G′) + γLℓR(C)− 1.
Case 2: |NC(x)∩ D| > ℓ. For easy writing, set r = |NC(x)∩ D|. Then D ∩ C is an LrR-dominating set of C. Hence |D ∩ C| ≥ γLrR(C). Let D = (D− C) ∪ DC ∪ Y , where Y is a subset of NG′(x) of cardinality r− ℓ.
Note that Y may contain vertices that are already in D. Clearly|DC∪ Y | = γLℓR(C) + r− ℓ. We now show that |D ∩ C| ≥ |DC ∪ Y |. Since r > ℓ ≥ s, by Proposition 4, γLrR(C) = γLℓR(C) + r− ℓ and thus |D ∩ C| ≥ γLrR(C) = γLℓR(C) + r− ℓ = |DC ∪ Y | holds. Consequently, we have |D| ≥ |DC|.
By the choice of D, x is included in D. Also that NC(x)∩ D ≥ ℓ and thus D∩ C is an LℓR-dominating set of C. Now we shall show that D∩ G′ is an L′-dominating set of G′. From the definition of ℓ and t, if t = 0 then L2 − ℓ = L2 − s ≤ degG′(x); if t > 0 then L2 − ℓ = degG′(x). In both cases, we have L2− ℓ ≤ degG′(x). Since D contains at least L2(x) vertices of NG(x), |NG′(x)∩ D| ≥ L2(x)− r. Let A be a vertex subset of NG′(x)∩ D of cardinality L2(x)− r. Now we pick arbitrarily exactly r − ℓ vertices from NG′(x)\ A to form the vertex set Y . Note that Y is well-defined as we have shown that L2− ℓ ≤ degG′(x). Since NG′(x)∩ D = (NG′(x)∩ D) ∪ Y , we have NG′(x)∩ D ≥ |A| + |Y | = L2(x)− ℓ = L′2(x). Consequently, γL(G) =|D| ≥ D = D∩ G′ + D∩ C −1≥ γL′(G′) + γLℓR(C)− 1.
Now we assume that L1(x) = B. If x has a neighbor y such that L2(y) = deg(y), then x must be included in any L-dominating set of G. Let L be the same as L except for the modification on L1(x) = R. In this case, we have γL(G) = γL(G). Thus the assertion holds clearly.
The argument of (3.2) is similar to (3.1) and therefore is omitted.
3.2. Algorithm
We are now in the position to present our algorithm, called kTTDom, to determine a minimum L-dominating set of a graph. In our algorithm, we assume kTTDomB is a subroutine that can find a minimum L-dominating set of each end-block C of a graph.
Algorithm: kTTDom (A block-wise approach for finding a γL-set in graphs. kTTDomB is a subroutine we assume it can find a γL-set of each end-block of the graph)
Input: A graph G with a proper labeling 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 ∪ kTTDomB(G′, L);
G′ ← ∅;
else
let C be an end-block of G′ and x be its unique cut-vertex;
α← min { L2(x), degC(x)};
U0 ← kTTDomB(C, L0);
UR← kTTDomB(C, LαR);
if |U0| = |UR| then D← D ∪ UR; L1(x)← R;
L2(x)← L2(x)− α;
else
if L1(x) = B and ∃ y ∈ NG(x) s.t. L2(y) = degG(y) then L1(x)← R;
s ← |NC(x)∩ Dx∗(C, L0)|;
ℓ ← s + max{L2(x)− s − degG′(x), 0};
D← D ∪ kTTDomB(C, Lℓ);
L2(x)← L2(x)− ℓ;
G′ ← (G′− C) ∪ { x };
end
Theorem 5. Algorithm kTTDom finds a minimum L-dominating set of a graph G in linear-time if kTTDomB and Dx∗(C, L0) take linear time to compute for each end-block C of G with cut-vertex x.
Proof. The correctness comes from Theorem 1. For the time complexity, since kTTDom calls at most three times of kTTDomB and computes D∗x(C, L0) at most once for each end-block C of the graph, it is clear that kTTDom is linear by the assumptions.
4. L-domination for some classes of graphs
In this section we present linear-time algorithms for finding a minimum L-dominating set for complete graphs, cycles and complete bipartite graphs.
In addition, the computations of D∗v(G, L) for the mentioned classes of graphs are also discussed. Combing with the algorithm presented in the previous section, we obtain a linear-time algorithm for the k-tuple total domination problem in graphs whose blocks are cliques, cycles or complete bipartite graphs. These include block graphs, cacti and block-cactus graphs.
Throughout the rest of this section, suppose G is a graph with a proper labeling L = (L1, L2). The aim is to find a minimum L-dominating set D of G. Define
R =e { v ∈ V : L1(v) = R or ∃ u ∈ NG(v) s.t. L2(u) = degG(u)} . (2) And let| eR| = r. By the definition of L-dominating set, all vertices of eR must be included in D.
4.1. Complete graphs
Suppose G = (V, E) is a complete graph with n vertices. Let V \ eR = { v1, v2, . . . , vn−r}. If vi ̸∈ D and vj ∈ D with L2(vi) < L2(vj) for some vi, vj ∈ V \ eR, then (D − vj) ∪ { vi} is also a minimum L-dominating set of G. This indicates that we shall choose vertices in V \ eR with L2 value as small as possible. Suppose vmaxR (resp. vmax) has the maximum L2 value among all vertices in eR (resp. V \ eR). It is the case that D = Re ∪ S, where S consists of the smallest s vertices of V \ eR, where s = max{ L2(vmaxR )− r + 1, L2(vmax)− r, 0 }. However, some vertex in S may have L2 value no less than r + s. In that case, one can simply add another vertex of V \ eR to S. In other words, choose the smallest s + 1 vertices of
V \ eR and eR to form a minimum L-dominating set of G. The process of the algorithm is described as follows.
Algorithm: kTTDomKn (Finding a γL-set of a complete graph)
Input: A complete graph G = (V, E) with a proper labeling L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
Re← { v ∈ V : L1(v) = R or ∃ u ∈ NG(v) s.t. L2(u) = degG(u)};
r ← | eR|;
v0 ← ∅ ; // pseudo vertex L2(v0)← 0;
let v1, v2, . . . , vn be a vertex ordering of V such that vi ∈ V \ eR for 1≤ i ≤ n − r, L2(v1)≤ · · · ≤ L2(vn−r) and L2(vn−r+1)≤ · · · ≤ L2(vn);
s ← max { L2(vn)− r + 1, L2(vn−r)− r, 0 };
if L2(vs)≥ r + s then s← s + 1;
D← eR∪ { vi ∈ V \ eR : 0≤ i ≤ s };
Theorem 6. Algorithm kTTDomKn 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 L is a proper labeling, 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, to obtain the vertex ordering.
Consider the computation of Dx∗(G, L) of a complete graph 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 |NG(x)∩ D| is maximum, and can be selected as D∗x(G, L). Thus D∗x(G, L) can be found in linear time.
4.2. Trees
Since each end-block of a tree T of order at least 2 is a clique of cardinality 2, a solution to find a minimum L-dominating set of a tree T is to use kTTDom and kTTDomKn. However, since tree has a simple structure,
we provide an alternative algorithm to find a minimum L-dominating set in trees. The presented algorithm is actually the simplified version of the combination of kTTDom and kTTDomKn for the input graph to be trees, and 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 breadth- first-search (BFS) algorithm.
The algorithm visits vertices of T along the tree ordering v1, v2, . . . , vn. In each iteration, processing a leaf vi of a tree T , which is adjacent to a unique vertex u. The label of vi is used to possibly relabel u. After v is visited, vi is ignored from T and a new tree T′ is obtained. A linear-time labeling algorithm for finding a γL-set in trees is shown as follows.
Algorithm: kTTDomT (Finding a γL-set of a tree)
Input: A tree T of n vertices with a tree ordering v1, v2, . . . , vn and a proper labeling L = (L1, L2).
Output: A minimum L-dominating set D of T . Method:
D← ∅;
for i ← 1 to n do T′ ← T [vi, . . . , vn];
let u be the parent of vi in T′ (regard u as vi if i = n);
if L2(vi) = 1 then L1(u)← R;
if L1(vi) = R or L2(u) = degT′(u) then // vi ∈ eR L2(u)← max { L2(u)− 1, 0 };
D← D ∪ { vi};
end
The construction and correctness of the algorithm is based on the follow- ing lemma.
Lemma 7. Suppose T is a tree of order at least 2 with a proper labeling L = (L1, L2). Let v be a leaf adjacent to u in T
(1) If L2(v) = 1, then γL(T ) = γL′(T ), where L′ is the same as L except for the modification on L′1(u) = R.
(2) If v ∈ eR, then γL(T ) = γL′(T − v) + 1, where L′ is the restriction of L on T − v with the modification on L′2(u) = max{ L2(u)− 1, 0 }
(3) If L(v) = (B, 0) and v ̸∈ eR, then γL(T ) = γL′(T − v), where L′ is the restriction of L on T − v.
Proof. (1) By the definition of L-domination, γL(T ) ≤ γL′(T ) holds clearly.
Now suppose D is a γL-set of T . Since L2(v) = 1, u must be included in D.
Thus D is also an L′-dominating set of T . Hence we have γL′(T ) ≤ γL(T ).
(2) By the Case (1), we can assume L1(u) = R if L2(v) = 1 without loss of generality. Suppose D′ is a γL′-set of T − v. Set D = D′ ∪ { v }.
Since L′ is the restriction of L on T − v with the modifications on L′2(u) and L2(u) ≤ L′2(u) + 1, we have |NT(u)∩ D| ≥ L2(u). If L2(v) = 1, then by assumption that L1(u) = R and thus u ∈ D′. Therefore D is also an L-dominating set of T . Thus γL(T )≤ |D| = |D′| + 1 = γL′(T − v) + 1.
Conversely, suppose D is a γL-set of T . By the assumption that v ∈ eR, v must be included in D. Set D′ = D\ { v }. As L′ is the restriction of L on T − v with the modification on L′2(u),|NT−v(x)∩ D′| ≥ L2(u)− 1 = L′2(u).
Thus D′ is an L′-dominating set of V (T′). Hence γL′(T− v) + 1 ≤ |D′| + 1 =
|D| = γL(T ).
(3) Suppose D′ is a γL′-set of T − v. Since L′ is the restriction of L on T − v and |N(v) ∩ D′| ≥ 0 = L2(v), it is clear that D′ is an L-dominating set of T . Thus γL(T )≤ |D′| = γL′(T − v).
Conversely, suppose D is a γL-set of T . If v ̸∈ D, then D is also an L′-dominating set of T − v. Now suppose v ∈ D. If |NT−v(u)∩ (D − v)| ≥ L′2(u), then we are done. For otherwise, pick a vertex w ∈ NT−v(u)\ D and set D′ = (D− v) ∪ { w }. Since v ̸∈ eR, we have L2(u) < degT(u) and thus
|NT−v(u)∩ D′| ≥ L′2(u). Therefore D′ is also an L′-dominating set of T − v.
Hence γL′(T − v) ≤ |D′| = |D| = γL(T ).
Theorem 8. Algorithm kTTDomT finds a minimum L-dominating set for a tree in linear-time.
4.3. Cycles
Suppose G = (V, E) is a cycle. We will use kTTDomT as a subroutine to find a minimum L-dominating set of G. Basically, the idea is to pick a particular vertex v∗of G, cut the cycle at v∗to get a path Pv∗, and then apply kTTDomT to obtain a minimum L′-dominating set of Pv∗. The principle of picking is to choose a vertex that must be included in D and is shown as
follows. If eR ̸= ∅, then we know that all vertices of eR must be included in D. Thus pick arbitrarily a vertex from eR as v∗. Now assume eR = ∅, i.e., L1(v) = B and L2(v) < 2 for all v ∈ V . If L2(v) = 0 for all v ∈ V , then clearly D =∅. Otherwise there must exist some vertex v that has L2(v) = 1.
This indicates that at least one of the two neighbors of v, say u and w, must be included in D. In this case, we can pick v∗ from u or w by testing the cardinality of the minimum L′-dominating set of C, where L′ is the same as L except for the modification on L1(u) = R or L1(w) = R.
After picking v∗, we need to modify the labels of its neighbors accordingly.
Let u∗ and w∗ be the two neighbors of v∗ on C. Since v∗ must be included in D, the L2 values of u∗ and w∗ should be decreased by 1. Let Pv∗ be the path of C − v∗ and LP be the restriction of L on Pv∗ with the modifications as described below.
Case 1: L2(v∗) = 2. Clearly u∗ and w∗ must be included in D. Thus just set the LP1 values of u∗ and w∗ as R.
Case 2: L2(v∗) = 1. So at least one of u∗ and w∗ must be included in D. In this case, we test the cardinality of the minimum LQ-dominating set of Pv∗ to decide which one of u∗ and w∗ should be chosen, where LQ is the same as LP except for the modifications on LQ1(u∗) = R or LQ1(w∗) = R.
Case 3: L2(v∗) = 0. Then none has to be changed in this case. The detailed algorithm is shown as in Algorithm kTTDomCYC.
The correctness is clear and therefore is omitted. The time complexity is clearly linear, since kTTDomT is linear and kTTDomCYC calls at most two times of kTTDomT.
Theorem 9. Algorithm kTTDomCYC finds a minimum L-dominating set in a cycle in linear time.
Now consider the computation of Dx∗(G, L) of a cycle G for some fixed vertex x. Let y and z be the neighbors of x in G. By the definition of D∗x(G, L), |NG(x)∩ D∗x(G, L)| ≤ 2. If |NG(x)∩ D∗x(G, L)| = 2, then y, z ∈ D∗x(G, L) and |Dx∗(G, L)| = γL′(G), where L′ is the same as L with the modifications on L1(y) = R and L1(z) = R. If |NG(x)∩ D∗x(G, L)| = 1 and suppose y ∈ D∗x(G, L), then |D∗x(G, L)| = γL′(G), where L′ is the same as L with the modifications on L1(y) = R. Thus one can find D∗x(G, L) by examining among all possible combinations of modifications on L′1(v) = R for all neighbors v of x with the condition that γL′(G) = γL(G). Since finding a γL-set of a cycle can be done in linear-time, the computation of D∗x(G, L) clearly can be done in linear time, too.
Algorithm: kTTDomCYC (Finding a γL-set of a cycle) Input: A cycle G = (V, E) with a proper labeling L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
Re← { v ∈ V : L1(v) = R or ∃ u ∈ NG(v) s.t. L2(u) = 2};
if eR̸= ∅ then
pick an arbitrarily vertex in eR as v∗; else if ∃ v s.t. L2(v) = 1 then
let u, w be the two neighbors of v;
Uu ← kTTDomCYC(G, L′), where L′ ← L with L′1(u)← R;
Uw ← kTTDomCYC(G, L′), where L′ ← L with L′1(w)← R;
if |Uu| ≤ |Uw| then D ← Uu else D ← Uw; else stop ;
let the two neighbors of v∗ be u∗ and w∗;
let Pv∗ be the path G− v∗ and LP be the restriction of L on Pv∗; LP2(u∗)← max{L2(u∗)− 1, 0};
LP2(w∗)← max{L2(w∗)− 1, 0};
if L2(v∗) = 2 then
LP1(u∗)← LP1(w∗)← R;
D← kTTDomT(Pv∗, LP)∪ {v∗};
else if L2(v∗) = 1 then
Uu∗ ← kTTDomT(Pv∗, LQ), where LQ ← LP with LQ1(u∗)← R;
Uw∗ ← kTTDomT(Pv∗, LQ), where LQ ← LP with LQ1(w∗)← R;
if |Uu∗| ≤ |Uw∗| then D ← Uu∗∪ {v∗} else D ← Uw∗ ∪ {v∗};
else
D← kTTDomT(Pv∗, LP)∪ {v∗};
4.4. Complete bipartite graphs
Suppose G = (A∪ B, E) is a complete bipartite graph whose vertex set is a disjoint union of two independent sets A and B. Let r1 = |A ∩ eR| and r2 =|B ∩ eR|. The argument is similar to that of complete graphs. Let amax
(resp. bmax) be a vertex in A that has the maximum L2 value among all vertices of A (resp. B). Since D must contain at least L2(amax) vertices of B, it is the case that we shall choose the smallest max{L2(amax)− r2, 0} (resp. max{L2(bmax)− r1, 0}) vertices of B \ eR (resp. A\ eR). The process of the algorithm is described as follows.
Algorithm kTTDomKmn (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 proper labeling L = (L1, L2).
Output: A minimum L-dominating set D of G.
Method:
D← ∅;
Re← { v ∈ V : L1(v) = R or ∃ u ∈ NG(v) s.t. L2(u) = degG(u)};
r1 ← |A ∩ eR|;
r2 ← |B ∩ eR|;
let a1, a2, . . . , a|A| and b1, b2, . . . , b|B| be vertex orderings of A and B, respectively, such that ai ∈ A \ eR for 1≤ i ≤ |A| − r1, bi ∈ B \ eR for 1≤ i ≤ |B| − r2, L2(a1)≤ · · · ≤ L2(a|A|−r1), L2(a|A|−r1+1)≤ · · · ≤ L2(a|A|), L2(b1)≤ · · · ≤ L2(b|B|−r2), and L2(b|B|−r2+1)≤ · · · ≤ L2(b|B|);
a0 ← ∅; b0 ← ∅ ; // pseudo vertices sa← max{L2(a|A|)− r2, L2(a|A|−r1)− r2, 0};
sb ← max{L2(a|B|)− r1, L2(a|B|−r1)− r1, 0};
D← eR∪ { ai ∈ A \ eR : 0≤ i ≤ sa} ∪ { bi ∈ B \ eR : 0≤ i ≤ sb};
Theorem 10. Algorithm kTTDomKmn 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.
The computation of D∗x(G, L) of a complete bipartite graph G for some fixed vertex x can be found in linear time as any minimum L-dominating set D of G has the property that |NG(x)∩ D| is maximum, and can be selected as Dx∗(G, L).
It is well-known that block graphs, cacti, block-cactus graphs can be recognized in linear-time. By Theorems 5, 6, 9, and 10, one can immediately have the following result.
Theorem 11. Algorithm kTTDom finds a minimum L-dominating set in linear 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.
5. NP-completeness result
In this section, we study the complexity of the k-tuple total domination problem:
k-TUPLE TOTAL DOMINATION (kTTD)
INSTANCE: A graph G = (V, E) and positive integers k and s.
QUESTION: Does G have a k-tuple total dominating set of size ≤ s?
It has been proved that kTTD is NP-complete for bipartite graphs and split graphs [14], in which the reductions are mainly from the well-known vertex cover problem. Pradhan also showed that kTTD is NP-complete for doubly chordal graphs, a subclass of chordal graphs. In this section, we show that kTTD remains NP-complete for undirected path graphs, another sub- class of chordal graphs. The argument is similar to that in [12], where the re- duction is from another well-known NP-complete problem, the 3-dimensional matching problem. For the sake of completeness, we will describe the reduc- tion completely.
3-DIMENSIONAL MATCHING (3DM)
INSTANCE: Disjoint sets X, Y and Z, each of cardinality q, and a set M ⊆ X× Y × Z of triples having cardinality p.
QUESTION: Is there a set of q triples in M such that each element of X ∪ Y ∪ Z is contained in exactly one of these triples?
Theorem 12. For any fixed positive integer k, kTTD is NP-complete for undirected path graphs.
Proof. Obviously kTTD belongs to NP, since it is easy to verify a “yes”
instance of kTTD in polynomial time. Consider an instance of 3DM. Let X ={ xr: 1≤ r ≤ q } , Y = { ys: 1≤ s ≤ q } , Z = { zt: 1≤ t ≤ q } , and a subset
M ={ mi = (xr, ys, zt) : xr ∈ X, ys ∈ Y and zt∈ Z for 1 ≤ i ≤ p } of triples X× Y × Z. Now we construct a clique tree T having 6p + 3q + 1 cliques from which we will obtain an undirected path graph. The vertices of the tree T , which are represented by sets, are explained below.
