**PREPRINT**

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

### ~ mathlib/preprint/2013- 10.pdf

## 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

^{I}

James K. Lan^{a,}* ^{∗}*, Gerard Jennhwa Chang

^{a,b,c}

*a**Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan*

*b**Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan*

*c**National 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 γ*_{t}*(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 N*_{G}*(v) =* *{ u ∈ V : uv ∈ E } and*
*the closed neighborhood is N**G**[v] = N**G**(v)∪ { v }. The degree deg**G**(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) = (L*_{1}*(v), L*_{2}*(v)), where L*_{1}*(v)* *∈ { B, R }, and L*2*(v) is*
*a nonnegative integer. Here a vertex v with L*_{1}*(v) = R is called a required*
*vertex ; a vertex v with L*_{1}*(v) = B is called a bound vertex. An L-dominating*
*set of G is a subset D* *⊆ V (G) such that*

*• if L*1*(v) = R, then v* *∈ D, and*

*• for each v ∈ V (G), |N**G**(v)∩ D| ≥ L*2*(v).*

*That is, D contains all required vertices, and for each vertex v of G, v is*
*adjacent to at least L*_{2}*(v) vertices in D. The L-domination number γ*_{L}*(G)*
*is the minimum cardinality of an L-dominating set in G, such set is called a*
*γ*_{L}*-set of G. Clearly G has an L-dominating set if and only if L*_{2}*(v)≤ deg**G**(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 γ**L**(G) = γ*_{×k,t}*(G). Thus an algorithm for γ**L**(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 = (L*1*, L*2*). 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*

*≤ deg*

*C*

*(x). The following notations will be*used throughout the rest of this section.

*• L*^{ℓ}*: the restriction of L on C with the modiﬁcation on L*^{ℓ}_{2}*(x) = ℓ.*

*• L*^{ℓR}*: the same as L*^{ℓ}*except for the modiﬁcation on L*^{ℓR}_{1} *(x) = R.*

*Since the value L*_{2}*(x) may be greater than the number of neighbors of x in C,*
*we need to set the L*2 *value of x as the minimum between L*2*(x) and deg*_{C}*(x)*
*before evaluating the cardinality of a γ*_{L}*-set of C. Thus, for convenience, set*

*α = min{L*2*(x), deg*_{C}*(x)}.*

*Note that one always have γ*_{L}^{0}*(C)* *≤ γ**L*^{ℓ}*(C)* *≤ γ**L*^{ℓR}*(C). Especially, substi-*
*tuting α for ℓ gives*

*γ*_{L}^{0}*(C)≤ γ**L*^{α}*(C)≤ γ**L*^{αR}*(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.,* *|N**G**(v)∩ D**v*^{∗}*(G, L)| ≥*

*|N**G**(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 D*_{v}^{∗}*(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 = (L**_{1}*, L*_{2}*).*

*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{L*2

*(x), deg*

_{C}*(x)}, s = |N*

*C*

*(x)∩ D*

^{∗}*x*

*(C, L*

^{0})

*| and*

*t = max{L*2

*(x)− s − deg*

*G*

^{′}*(x), 0}.*

*(1) If γ*_{L}^{0}*(C) = γ*_{L}^{αR}*(C), then γ*_{L}*(G) = γ*_{L}*′**(G*^{′}*)+γ*_{L}^{αR}*(C)−1, where L** ^{′}*1

*(x) =*

*R, L*

^{′}_{2}

*(x) = L*

_{2}

*(x)− α.*

*(2) If γ*_{L}^{0}*(C) = γ**L*^{α}*(C) < γ*_{L}^{αR}*(C), then γ**L**(G) = γ**L*^{′}*(G*^{′}*) + γ**L*^{α}*(C), where*
*L*^{′}_{2}*(x) = L*_{2}*(x)− α.*

*(3) Suppose γ*_{L}^{0}*(C) < γ*_{L}^{α}*(C).*

*(3.1) If L*_{1}*(x) = R or there exists y* *∈ N**G**(x) such that L*_{2}*(y) = deg*_{G}*(y),*
*then γ*_{L}*(G) = γ*_{L}*′**(G*^{′}*) + γ*_{L}*ℓR**(C)− 1, where L** ^{′}*1

*(x) = R, L*

^{′}_{2}

*(x) =*

*L*

_{2}

*(x)− ℓ and ℓ = s + t.*

*(3.2) if L*_{2}*(y) < deg*_{G}*(y) for all y* *∈ N**G**(x), then γ*_{L}*(G) = γ*_{L}*′**(G** ^{′}*) +

*γ*

_{L}*ℓ*

*(C), where L*

^{′}_{2}

*(x) = L*

_{2}

*(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***γ*_{L}*-set of G. Then* *|D ∩ C| ≥ γ**L*^{0}*(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 γ**L**-set of G. Let α = min{L*2*(x), deg*_{C}*(x)}. Let D**C* *be a γ**L*^{α}*-set or a*
*γ*_{L}^{αR}*-set of C and D = (D− C) ∪ D**C**. Then* *N*_{G}*′**(x)∩ D ≥L*_{2}*(x)− α and*
*N*_{C}*(x)∩ D ≥α. In other words,* *N*_{G}*(x)∩ D ≥L*_{2}*(x).*

*Proof. By the deﬁnition of L-dominating set,* *|N**G**(x)∩ D| ≥ L*2*(x) and*
*N*_{C}*(x)∩ D ≥α. If L*_{2}*(x)≤ deg**C**(x), i.e., α = L*_{2}*(x), then* *N*_{G}*′**(x)∩ D ≥*
*0 = L*_{2}*(x)− α. If L*2*(x) > deg*_{C}*(x), i.e., α = deg*_{C}*(x), then α = deg*_{C}*(x)≥*

*|N**C**(x)∩ D|, as x can have at most deg**C**(x) neighbors in D. Hence|N**G*^{′}*(x)∩*
*D| = |N**G*^{′}*(x)∩ D| = |N**G**(x)∩ D| − |N**C**(x)∩ D| ≥ L*2*(x)− α. The assertion*
holds clearly.

**Proof of Theorem 1 (1). Let D**^{′}*be a γ*_{L}*′**-set of G*^{′}*and D*_{C}*be a γ*_{L}* ^{αR}*-set of

*C. Since L*

^{′}_{1}

*(x) = L*

^{αR}_{1}

*(x) = R, x∈ D*

^{′}*∪D*

*C*. Also that

*|N*

*G*

*(x)∩ (D*

^{′}*∪ D*

*C*)

*| =*

*|N**G**(x)∩ D*^{′}*| + |N**G**(x)∩ D**C**| ≥ L** ^{′}*2

*(x) + α = L*2

*(x), by the assumption. We*

*have D*

^{′}*∪ D*

*C*

*is an L-dominating set of G and hence γ*

_{L}*(G)*

*≤ |D*

^{′}*∪ D*

*C*

*| =*

*|D*^{′}*| + |D**C**| − 1 = γ**L*^{′}*(G*^{′}*) + γ*_{L}^{αR}*(C)− 1.*

*Conversely, suppose D is a γ*_{L}*-set of G. Let D*_{C}*be a γ*_{L}^{αR}*-set of C and let*
*D = (D− C) ∪ D**C**. By the assumption that γ*_{L}^{0}*(C) = γ*_{L}^{αR}*(C) and Eq. (1)*
and Proposition 2,*|D| ≥**D**. Note that no matter what L*1*(x) might be, x is*
*always be included in D. By Lemma 3,* *N*_{G}*(x)∩ D ≥L*_{2}*(x) and thus D is*
*also an L-dominating set of G. Moreover,**N*_{G}*′**(x)∩ D ≥L*_{2}*(x)−α = L** ^{′}*2

*(x)*and N

_{C}*(x)∩ D ≥α. Thus D∩ G*

^{′}*is an L*

^{′}*-dominating set of G*

^{′}*and D∩ C*

*is an L*

^{αR}*-dominating set of C. Hence γ*

_{L}*(G) =*

*|D| ≥*

*D*=

*D∩ G*

*+*

^{′}*D∩ C −*1

*≥ γ*

*L*

^{′}*(G*

^{′}*) + γ*

_{L}

^{αR}*(C)− 1.*

*Note that under the assumptions of Theorem 1 (2), x must be a bound*
*vertex, i.e., L*_{1}*(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 =* *|N**G**(x)∩ D*^{∗}*x**(G, L*^{0})*|*
*and j be a nonnegative integer such that s + j* *≤ deg**G**(x). Then γ*_{L}^{s+j}*(G) =*
*γ*_{L}^{s}*(G) + j.*

*Proof. Let D*_{s}*be a γ*_{L}^{s}*-set of G and D = D*_{s}*∪J, where J consists of j vertices*
*of N*_{G}*(x)\ D**s**. Clearly D is an L*^{s+j}*-dominating set of G and γ*_{L}^{s+j}*(G)* *≤*

*|D| = |D**s**| + j = γ**L*^{s}*(G) + j holds.*

*Conversely, suppose D is a γ*_{L}^{s+j}*-set of G and we have* *|N**G**(x)∩ D| ≥*
*s + j.* *Since D must contain an L*^{0}*-dominating set of G, there exists a*
*subset J* *⊆ V (G) such that D \ J is an L*^{0}*-dominating set of G. Clearly,*

*|J| ≤ γ**L*^{s+j}*(G)− γ**L*^{0}*(G). Also, by the deﬁnition of s, γ*_{L}^{0}*(G) = γ*_{L}^{s}*(G) and*

*|N**G**(x)∩ (D \ J)| ≤ s. This indicates that |J| ≥ j. Therefore, γ**L*^{s+j}*(G)* *≥*
*γ*_{L}^{0}*(G) + j = γ*_{L}^{s}*(G) + j.*

**Proof of Theorem 1 (3). We ﬁrst consider (3.1) and assume that L**_{1}*(x) =*
*R. It is easy to check that L*^{ℓR}*is a proper labeling on V (C). Let D** ^{′}* be a

*γ*

*L*

^{′}*-set of G*

^{′}*and D*

*C*

*be a γ*

_{L}

^{ℓR}*-set of C. Since L*

^{′}_{1}

*(x) = L*

^{ℓR}_{1}

*(x) = R, x∈ D*

^{′}*∪*

*D*

*. Also by assumptions, we have*

_{C}*|N*

*G*

*(x)∩ (D*

^{′}*∪ D*

*C*)

*| = |N*

*G*

*(x)∩ D*

^{′}*| +*

*|N**G**(x)∩ D**C**| ≥ L*2*(x). Thus D*^{′}*∪D**C* *is an L-dominating set of G and hence*
*γ**L**(G)≤ |D*^{′}*∪ D**C**| = |D*^{′}*| + |D**C**| − 1 = γ**L*^{′}*(G*^{′}*) + γ*_{L}^{ℓR}*(C)− 1.*

*Conversely, suppose D is a γ*_{L}*-set of G. Let D*_{C}*be a γ*_{L}*ℓR**-set of C.*

*Case 1:* *|N**C**(x)∩ D| ≤ ℓ. Let D = (D − C) ∪ D**C*. We now claim that

*|D ∩ C| ≥ γ**L*^{ℓR}*(C). If this claim holds, then we have* *|D| ≥* *D*. First
*consider the case of t = 0. Since ℓ = s + t, we have γ*_{L}*ℓR**(C) = γ*_{L}^{sR}*(C).*

*Since x can have at most s neighbors in a γ*_{L}^{0R}*-set of C, γ*_{L}^{sR}*(C) = γ*_{L}^{0R}*(C).*

*By Proposition 2 and the assumption that L*_{1}*(x) = R,* *|D ∩ C| ≥ γ**L*^{0}*(C) =*
*γ*_{L}^{0R}*(C) = γ*_{L}^{ℓR}*(C) holds.*

*Now suppose t > 0. From the deﬁnition of t, we have L*2*(x) = ℓ+deg*_{G}*′**(x).*

*Since D contains at least L*_{2}*(x) vertices of N*_{G}*(x), D must contain at least ℓ*
*vertices of N*_{C}*(x), i.e.,|N**C**(x)∩ D| ≥ ℓ, for otherwise we have |N**G*^{′}*(x)∩ D| >*

*L*2*(x)− ℓ = deg**G*^{′}*(x), a contradiction. By the assumption that L*1*(x) = R,*
*D∩ C is also an L*^{ℓR}*-dominating set of C. Hence* *|D ∩ C| ≥ γ**L*^{ℓR}*(C). We*
therefore have this claim.

*By the choice of D, x is included in D. Also that* *N**C**(x)∩ D ≥* *ℓ*
*and thus D* *∩ C is an L*^{ℓR}*-dominating set of C. By the assumption that*

*|N**C**(x)∩ D| ≤ ℓ, we have* *N*_{G}*′**(x)∩ D* = *|N**G**(x)∩ D| − |N**C**(x)∩ D| ≥*
*L*_{2}*(x)− ℓ = L** ^{′}*2

*(x), and thus D∩ G*

^{′}*is also an L*

^{′}*-dominating set of G*

*. Hence*

^{′}*γ*

_{L}*(G) =|D| ≥*

*D*=

*D∩ G*

*+*

^{′}*D∩ C −*1

*≥ γ*

*L*

^{′}*(G*

^{′}*) + γ*

_{L}

^{ℓR}*(C)− 1.*

*Case 2:* *|N**C**(x)∩ D| > ℓ. For easy writing, set r = |N**C**(x)∩ D|. Then*
*D* *∩ C is an L*^{rR}*-dominating set of C.* Hence *|D ∩ C| ≥ γ**L*^{rR}*(C).* Let
*D = (D− C) ∪ D**C* *∪ Y , where Y is a subset of N**G*^{′}*(x) of cardinality r− ℓ.*

*Note that Y may contain vertices that are already in D. Clearly|D**C**∪ Y | =*
*γ*_{L}^{ℓR}*(C) + r− ℓ. We now show that |D ∩ C| ≥ |D**C* *∪ Y |. Since r > ℓ ≥ s,*
*by Proposition 4, γ*_{L}^{rR}*(C) = γ*_{L}^{ℓR}*(C) + r− ℓ and thus |D ∩ C| ≥ γ**L*^{rR}*(C) =*
*γ*_{L}*ℓR**(C) + r− ℓ = |D**C* *∪ Y | holds. Consequently, we have |D| ≥ |D**C**|.*

*By the choice of D, x is included in D. Also that* *N*_{C}*(x)∩ D ≥* *ℓ and*
*thus D∩ C is an L*^{ℓR}*-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*
*L*_{2} *− ℓ = L*2 *− s ≤ deg**G*^{′}*(x); if t > 0 then L*_{2} *− ℓ = deg**G*^{′}*(x). In both*
*cases, we have L*2*− ℓ ≤ deg**G*^{′}*(x). Since D contains at least L*2*(x) vertices of*
*N*_{G}*(x),* *|N**G*^{′}*(x)∩ D| ≥ L*2*(x)− r. Let A be a vertex subset of N**G*^{′}*(x)∩ D*
*of cardinality L*_{2}*(x)− r. Now we pick arbitrarily exactly r − ℓ vertices from*
*N**G*^{′}*(x)\ A to form the vertex set Y . Note that Y is well-deﬁned as we*
*have shown that L*_{2}*− ℓ ≤ deg**G*^{′}*(x). Since N*_{G}*′**(x)∩ D = (N**G*^{′}*(x)∩ D) ∪ Y ,*
we have *N*_{G}*′**(x)∩ D ≥ |A| + |Y | = L*2*(x)− ℓ = L** ^{′}*2

*(x). Consequently,*

*γ*

*L*

*(G) =|D| ≥*

*D*=

*D∩ G*

*+*

^{′}*D∩ C −*1

*≥ γ*

*L*

^{′}*(G*

^{′}*) + γ*

_{L}

^{ℓR}*(C)− 1.*

*Now we assume that L*_{1}*(x) = B. If x has a neighbor y such that L*_{2}*(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 L*1*(x) = R. In this case, we have*
*γ*_{L}*(G) = γ*_{L}*(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 γ*** _{L}*-set in

**graphs. kTTDomB is a subroutine we assume it can ﬁnd a**

*γ*

*L*-set of each end-block of the graph)

**Input: A graph G with a proper labeling L = (L**_{1}*, L*_{2}).

**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 { L*2*(x), deg*_{C}*(x)};*

*U*_{0} **← kTTDomB(C, L**^{0});

*U*_{R}**← kTTDomB(C, L*** ^{αR}*);

**if** *|U*0*| = |U**R***| then***D← D ∪ U**R*;
*L*_{1}*(x)← R;*

*L*_{2}*(x)← L*2*(x)− α;*

**else**

**if L**_{1}*(x) = B and* *∃ y ∈ N**G**(x) s.t. L*_{2}*(y) = deg*_{G}**(y) then***L*1*(x)← R;*

*s* *← |N**C**(x)∩ D**x*^{∗}*(C, L*^{0})*|;*

*ℓ* *← s + max{L*2*(x)− s − deg**G*^{′}*(x), 0};*

*D ← D ∪ kTTDomB(C, L*

*);*

^{ℓ}*L*_{2}*(x)← L*2*(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 D**_{x}^{∗}*(C, L*^{0}*) 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, L*^{0}*) 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 = (L*_{1}*, L*_{2}*). The aim is to ﬁnd a minimum L-dominating set D of*
*G. Deﬁne*

*R =*e *{ v ∈ V : L*1*(v) = R or* *∃ u ∈ N**G**(v) s.t. L*_{2}*(u) = deg*_{G}*(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 =*
*{ v*1*, v*_{2}*, . . . , v*_{n}_{−r}*}. If v**i* *̸∈ D and v**j* *∈ D with L*2*(v*_{i}*) < L*_{2}*(v** _{j}*) for some

*v*

_{i}*, v*

_{j}*∈ V \ eR, then (D*

*− v*

*j*)

*∪ { v*

*i*

*} is also a minimum L-dominating*

*set of G. This indicates that we shall choose vertices in V*

*\ eR with L*

_{2}

*value as small as possible. Suppose v*

_{max}

^{R}*(resp. v*

*) has the maximum*

_{max}*L*2 value among all vertices in e

*R (resp. V*

*\ eR). It is the case that D =*

*R*e

*∪ S, where S consists of the smallest s vertices of V \ eR, where s =*max

*{ L*2

*(v*

_{max}*)*

^{R}*− r + 1, L*2

*(v*

*)*

_{max}*− r, 0 }. However, some vertex in S may*

*have L*

_{2}

*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 γ*** _{L}*-set of a complete graph)

**Input: A complete graph G = (V, E) with a proper labeling L = (L**_{1}*, L*_{2}).

**Output: A minimum L-dominating set D of G.**

**Method:**

*D← ∅;*

*R*e*← { v ∈ V : L*1*(v) = R or* *∃ u ∈ N**G**(v) s.t. L*_{2}*(u) = deg*_{G}*(u)};*

*r* *← | eR|;*

*v*_{0} *← ∅ ; // pseudo vertex*
*L*_{2}*(v*_{0})*← 0;*

*let v*_{1}*, v*_{2}*, . . . , v*_{n}*be a vertex ordering of V such that v*_{i}*∈ V \ eR for*
1*≤ i ≤ n − r, L*2*(v*_{1})*≤ · · · ≤ L*2*(v*_{n}_{−r}*) and L*_{2}*(v*_{n}* _{−r+1}*)

*≤ · · · ≤ L*2

*(v*

*);*

_{n}*s* *← max { L*2*(v**n*)*− r + 1, L*2*(v**n**−r*)*− r, 0 };*

**if L**_{2}*(v** _{s}*)

**≥ r + s then***s← s + 1;*

*D← eR∪ { v**i* *∈ 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, L*_{2}*(v)* *≤ deg**G**(v) < n for all v* *∈ V \ eR. Since each L*_{2}*(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 D*_{x}^{∗}*(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* *|N**G**(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*
*v*_{1}*, v*_{2}*, . . . , v*_{n}*such that v*_{i}*is a leaf of G*_{i}*= G[v*_{i}*, v*_{i+1}*, . . . , v** _{n}*] 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 v*_{1}*, v*_{2}*, . . . , v** _{n}*. In

*each iteration, processing a leaf v*

_{i}*of a tree T , which is adjacent to a unique*

*vertex u. The label of v*

*i*

*is used to possibly relabel u. After v is visited,*

*v*

_{i}*is ignored from T and a new tree T*

*is obtained. A linear-time labeling*

^{′}*algorithm for ﬁnding a γ*

*-set in trees is shown as follows.*

_{L}**Algorithm: kTTDomT (Finding a γ*** _{L}*-set of a tree)

**Input: A tree T of n vertices with a tree ordering v**_{1}*, v*_{2}*, . . . , v** _{n}* and a

*proper labeling L = (L*

_{1}

*, L*

_{2}).

**Output: A minimum L-dominating set D of T .****Method:**

*D← ∅;*

**for i****← 1 to n do***T*^{′}*← T [v**i**, . . . , v** _{n}*];

*let u be the parent of v*_{i}*in T*^{′}*(regard u as v*_{i}*if i = n);*

**if L**_{2}*(v*_{i}**) = 1 then**
*L*_{1}*(u)← R;*

* if L*1

*(v*

*i*

*) = R or L*2

*(u) = deg*

_{T}*′*

**(u) then // v***i*

*∈ eR*

*L*

_{2}

*(u)← max { L*2

*(u)− 1, 0 };*

*D← D ∪ { v**i**};*

**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 = (L*_{1}*, L*_{2}*). Let v be a leaf adjacent to u in T*

*(1) If L*_{2}*(v) = 1, then γ*_{L}*(T ) = γ*_{L}*′**(T ), where L*^{′}*is the same as L except for*
*the modification on L*^{′}_{1}*(u) = R.*

*(2) If v* *∈ eR, then γ*_{L}*(T ) = γ*_{L}*′**(T* *− v) + 1, where L*^{′}*is the restriction of L*
*on T* *− v with the modification on L** ^{′}*2

*(u) = max{ L*2

*(u)− 1, 0 }*

*(3) If L(v) = (B, 0) and v* *̸∈ eR, then γ*_{L}*(T ) = γ*_{L}*′**(T* *− v), where L*^{′}*is the*
*restriction of L on T* *− v.*

*Proof. (1) By the deﬁnition of L-domination, γ*_{L}*(T )* *≤ γ**L*^{′}*(T ) holds clearly.*

*Now suppose D is a γ*_{L}*-set of T . Since L*_{2}*(v) = 1, u must be included in D.*

*Thus D is also an L*^{′}*-dominating set of T . Hence we have γ**L*^{′}*(T )* *≤ γ**L**(T ).*

*(2) By the Case (1), we can assume L*_{1}*(u) = R if L*_{2}*(v) = 1 without*
*loss of generality. Suppose D*^{′}*is a γ*_{L}*′**-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 L*

_{2}

*(u)*

*≤ L*

*2*

^{′}*(u) + 1, we have*

*|N*

*T*

*(u)∩ D| ≥ L*2

*(u). If L*

_{2}

*(v) = 1, then*

*by assumption that L*

_{1}

*(u) = R and thus u*

*∈ D*

^{′}*. Therefore D is also an*

*L-dominating set of T . Thus γ*

*L*

*(T )≤ |D| = |D*

^{′}*| + 1 = γ*

*L*

^{′}*(T*

*− v) + 1.*

*Conversely, suppose D is a γ*_{L}*-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),|N*

*T*

*−v*

*(x)∩ D*

^{′}*| ≥ L*2

*(u)− 1 = L*

*2*

^{′}*(u).*

*Thus D*^{′}*is an L*^{′}*-dominating set of V (T*^{′}*). Hence γ*_{L}*′**(T− v) + 1 ≤ |D*^{′}*| + 1 =*

*|D| = γ**L**(T ).*

*(3) Suppose D*^{′}*is a γ*_{L}*′**-set of T* *− v. Since L*^{′}*is the restriction of L on*
*T* *− v and |N(v) ∩ D*^{′}*| ≥ 0 = L*2*(v), it is clear that D*^{′}*is an L-dominating*
*set of T . Thus γ*_{L}*(T )≤ |D*^{′}*| = γ**L*^{′}*(T* *− v).*

*Conversely, suppose D is a γ*_{L}*-set of T . If v* *̸∈ D, then D is also an*
*L*^{′}*-dominating set of T* *− v. Now suppose v ∈ D. If |N**T**−v**(u)∩ (D − v)| ≥*
*L*^{′}_{2}*(u), then we are done. For otherwise, pick a vertex w* *∈ N**T**−v**(u)\ D and*
*set D*^{′}*= (D− v) ∪ { w }. Since v ̸∈ eR, we have L*_{2}*(u) < deg*_{T}*(u) and thus*

*|N**T**−v**(u)∩ D*^{′}*| ≥ L** ^{′}*2

*(u). Therefore D*

^{′}*is also an L*

^{′}*-dominating set of T*

*− v.*

*Hence γ**L*^{′}*(T* *− v) ≤ |D*^{′}*| = |D| = γ**L**(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 P*_{v}*∗*, and then apply
**kTTDomT to obtain a minimum L**^{′}*-dominating set of P**v** ^{∗}*. The principle

*of picking is to choose a vertex that must be included in D and is shown as*

follows. If e*R* *̸= ∅, then we know that all vertices of eR must be included in*
*D. Thus pick arbitrarily a vertex from eR as v** ^{∗}*. Now assume e

*R =*

*∅, i.e.,*

*L*1

*(v) = B and L*2

*(v) < 2 for all v*

*∈ V . If L*2

*(v) = 0 for all v*

*∈ V , then*

*clearly D =∅. Otherwise there must exist some vertex v that has L*2

*(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 L*

_{1}

*(u) = R or L*

_{1}

*(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 L*

_{2}

*values of u*

^{∗}*and w*

^{∗}*should be decreased by 1. Let P*

_{v}*∗*be the

*path of C*

*− v*

^{∗}*and L*

^{P}*be the restriction of L on P*

*v*

*with the modiﬁcations as described below.*

^{∗}*Case 1: L*_{2}*(v*^{∗}*) = 2. Clearly u*^{∗}*and w*^{∗}*must be included in D. Thus just*
*set the L*^{P}_{1} *values of u*^{∗}*and w*^{∗}*as R.*

*Case 2: L*_{2}*(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 L*

*-dominating set*

^{Q}*of P*

*v*

^{∗}*to decide which one of u*

^{∗}*and w*

^{∗}*should be chosen, where L*

*is the*

^{Q}*same as L*

^{P}*except for the modiﬁcations on L*

^{Q}_{1}

*(u*

^{∗}*) = R or L*

^{Q}_{1}

*(w*

^{∗}*) = R.*

*Case 3: L*_{2}*(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 D*_{x}^{∗}*(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),* *|N**G**(x)∩ D*^{∗}*x**(G, L)| ≤ 2. If |N**G**(x)∩ D*^{∗}*x**(G, L)| = 2, then y, z ∈*
*D*^{∗}_{x}*(G, L) and* *|D**x*^{∗}*(G, L)| = γ**L*^{′}*(G), where L*^{′}*is the same as L with the*
*modiﬁcations on L*_{1}*(y) = R and L*_{1}*(z) = R. If* *|N**G**(x)∩ D*^{∗}*x**(G, L)| = 1 and*
*suppose y* *∈ D*^{∗}*x**(G, L), then* *|D*^{∗}*x**(G, L)| = γ**L*^{′}*(G), where L** ^{′}* is the same as

*L with the modiﬁcations on L*1

*(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 γ*

_{L}*′*

*(G) = γ*

_{L}*(G). Since ﬁnding*

*a γ*

*L*

*-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 γ*** _{L}*-set of a cycle)

**Input: A cycle G = (V, E) with a proper labeling L = (L**_{1}

*, L*

_{2}).

**Output: A minimum L-dominating set D of G.**

**Method:**

*D← ∅;*

*R*e*← { v ∈ V : L*1*(v) = R or* *∃ u ∈ N**G**(v) s.t. L*_{2}*(u) = 2};*

**if e***R ̸= ∅ then*

pick an arbitrarily vertex in e*R as v** ^{∗}*;

**else if**

*∃ v s.t. L*2

**(v) = 1 then***let u, w be the two neighbors of v;*

*U*_{u}**← kTTDomCYC(G, L**^{′}*), where L*^{′}*← L with L** ^{′}*1

*(u)← R;*

*U*_{w}**← kTTDomCYC(G, L**^{′}*), where L*^{′}*← L with L** ^{′}*1

*(w)← R;*

**if** *|U**u**| ≤ |U**w***| then D ← U***u* **else D***← U**w*;
**else stop ;**

*let the two neighbors of v*^{∗}*be u*^{∗}*and w** ^{∗}*;

*let P**v*^{∗}*be the path G− v*^{∗}*and L*^{P}*be the restriction of L on P**v** ^{∗}*;

*L*

^{P}_{2}

*(u*

*)*

^{∗}*← max{L*2

*(u*

*)*

^{∗}*− 1, 0};*

*L*^{P}_{2}*(w** ^{∗}*)

*← max{L*2

*(w*

*)*

^{∗}*− 1, 0};*

* if L*2

*(v*

^{∗}**) = 2 then**

*L*^{P}_{1}*(u** ^{∗}*)

*← L*

*1*

^{P}*(w*

*)*

^{∗}*← R;*

*D ← kTTDomT(P*

*v*

^{∗}*, L*

*)*

^{P}*∪ {v*

^{∗}*};*

**else if L**_{2}*(v*^{∗}**) = 1 then**

*U*_{u}*∗* **← kTTDomT(P***v*^{∗}*, L*^{Q}*), where L*^{Q}*← L*^{P}*with L*^{Q}_{1}*(u** ^{∗}*)

*← R;*

*U*_{w}*∗* **← kTTDomT(P***v*^{∗}*, L*^{Q}*), where L*^{Q}*← L*^{P}*with L*^{Q}_{1}*(w** ^{∗}*)

*← R;*

**if** *|U**u*^{∗}*| ≤ |U**w*^{∗}**| then D ← U***u*^{∗}*∪ {v*^{∗}**} else D ← U***w*^{∗}*∪ {v*^{∗}*};*

**else**

*D ← kTTDomT(P*

*v*

^{∗}*, L*

*)*

^{P}*∪ {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 r*_{1} = *|A ∩ eR| and*
*r*_{2} =*|B ∩ eR|. The argument is similar to that of complete graphs. Let a**max*

*(resp. b*_{max}*) be a vertex in A that has the maximum L*_{2} value among all
*vertices of A (resp. B). Since D must contain at least L*_{2}*(a** _{max}*) vertices

*of B, it is the case that we shall choose the smallest max{L*2

*(a*

*)*

_{max}*− r*2

*, 0}*(resp. max

*{L*2

*(b*

*)*

_{max}*− r*1

*, 0}) vertices of B \ eR (resp. A\ eR). The process*of the algorithm is described as follows.

**Algorithm kTTDomKmn (Finding a γ*** _{L}*-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 = (L*_{1}*, L*_{2}).

**Output: A minimum L-dominating set D of G.**

**Method:**

*D← ∅;*

*R*e*← { v ∈ V : L*1*(v) = R or* *∃ u ∈ N**G**(v) s.t. L*_{2}*(u) = deg*_{G}*(u)};*

*r*_{1} *← |A ∩ eR|;*

*r*2 *← |B ∩ eR|;*

*let a*_{1}*, a*_{2}*, . . . , a*_{|A|}*and b*_{1}*, b*_{2}*, . . . , b*_{|B|}*be vertex orderings of A and B,*
*respectively, such that a*_{i}*∈ A \ eR for 1≤ i ≤ |A| − r*1*, b*_{i}*∈ B \ eR for*
1*≤ i ≤ |B| − r*2*, L*_{2}*(a*_{1})*≤ · · · ≤ L*2*(a*_{|A|−r}_{1}*), L*_{2}*(a*_{|A|−r}_{1}_{+1})*≤ · · · ≤ L*2*(a** _{|A|}*),

*L*

_{2}

*(b*

_{1})

*≤ · · · ≤ L*2

*(b*

_{|B|−r}_{2}

*), and L*

_{2}

*(b*

_{|B|−r}_{2}

_{+1})

*≤ · · · ≤ L*2

*(b*

*);*

_{|B|}*a*_{0} *← ∅; b*0 *← ∅ ; // pseudo vertices*
*s*_{a}*← max{L*2*(a** _{|A|}*)

*− r*2

*, L*

_{2}

*(a*

_{|A|−r}_{1})

*− r*2

*, 0};*

*s*_{b}*← max{L*2*(a** _{|B|}*)

*− r*1

*, L*

_{2}

*(a*

_{|B|−r}_{1})

*− r*1

*, 0};*

*D← eR∪ { a**i* *∈ A \ eR : 0≤ i ≤ s**a**} ∪ { b**i* *∈ B \ eR : 0≤ i ≤ s**b**};*

**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* *|N**G**(x)∩ D| is maximum, and can be selected*
*as D*_{x}^{∗}*(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 ={ x**r*: 1*≤ r ≤ q } , Y = { y**s*: 1*≤ s ≤ q } , Z = { z**t*: 1*≤ t ≤ q } ,*
and a subset

*M ={ m**i* *= (x*_{r}*, y*_{s}*, z*_{t}*) : x*_{r}*∈ X, y**s* *∈ Y and z**t**∈ 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.*