The Numerical Invariants concerning the Total Domination for Generalized Petersen Graphs

A subset S of V ( G ) is called a total dominating set of a graph G if every vertex in V ( G ) is adjacent to a vertex in S . The total domination number of a graph G denoted by c t ( G ) is the minimum cardinality of a total dominating set in G . The maximum order of a partition of V ( G ) into total dominating sets of G is called the total domatic number of G and is denoted by d t ( G ) . Domination in graphs has applications to several ﬁelds. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and ﬁre stations) is ﬁxed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.


Introduction
Graph theory has been used to study various concepts of partition of the vertex set for the graph G [1][2][3]. e domination (total domination) problem for the graph was studied from 1950s onward, and domination (total domination) in graphs is said to be NP-complete problem [4]. Domination (total domination) sets are of practical interest in several areas of mathematics and other branches of science [5,6]. In wireless networks, dominating (total dominating) sets are used to find efficient routes within ad hoc mobile networks. In documents' summarization, domination (total domination) sets are used. Dominating (total dominating) sets are closely related to independent sets. Total dominating set is independent if and only if it is a maximal input set [7].
Moreover, in computer science, weighted graphs are widely used in the developments of data mining, software testing, image processing, communication networks, and information security [8][9][10][11].

Preliminaries
Let G � (V, E) be a simple, undirected, and finite graph with the vertex set V(G) and the edge set E(G). e (open) neighborhood of a vertex u ∈ V(G), denoted by N(u), is the set of all vertices adjacent to u, i.e., [17][18][19]. For a set and its closed neighborhood is the set N[S] � N(S) ∪ S. A dominating set of a graph G, abbreviated as DS, is a set S of vertices in G such that every vertex in V(G)\S is adjacent to a vertex in S. e domination number of a graph G, denoted by c(G), is the minimum cardinality of a dominating set. A total dominating set of a graph G, abbreviated as TDS, is a set S of vertices in G such that every vertex in V(G) is adjacent to a vertex in S. If no proper subset of S is a TDS of G, then S is a minimal TDS of G. Every graph without isolated vertices has a TDS. e total domination number of a graph G, denoted by c t (G), is the minimum cardinality of a TDS. A TDS of G of minimum cardinality is called a c t (G)-set. e concept of total domination was introduced by Cockayne et al. [20]. For more information, see [21,22].
Let τ(G) be the total number of c t (G)-sets. For each vertex v ∈ V(G), we define the total domination value of v, denoted by TDV G (v), to be the number of c t (G)-sets to which v belongs. Evidently, 0 ≤ TDV G (v) ≤ τ(G) for any G and any v ∈ V(G).
Generalized Petersen graphs are important classes of commonly used interconnection networks and have been intensively studied. e generalized Petersen graph P(n, m), n ≥ 3 and 1 ≤ m ≤ ⌊(n − 1)/2⌋, consists of an outer n-cycle u 1 , u 2 , . . . , u n , a set of n spokes u i v i , 1 ≤ i ≤ n, and n edges v i v i+m , 1 ≤ i ≤ n, with indices taken as modulo n, see Figure 1.
us, its vertex set is the union of U � u 1 , u 2 , . . . , u n and V � v 1 , v 2 , . . . , v n . We say that the first set consists of u vertices and the second one of v vertices. By a u-path in P(n, m), we mean a path whose vertices consist of just u-vertices. A v-path is defined similarly. e generalized Petersen graph P(n, m) is shown in Figure 1.
Investigation of the total number of total domatic sets and the total number of total dominating sets of the generalized Petersen graph is the sole aim and interest of writing this paper. Cao et al. in [23] studied the total domination number of generalized Petersen graphs and obtained the exact value of c t (P(n, 2)). For k ≥ 2, they proved c t (P(n, 2)) � 2k, for n � 3k, 2k + 2, for n � 3k + 1, 2k + 2, for n � 3k + 2.
In this paper, we determine the exact values for c t (P(n, m)) when n ≡ 0 (mod 3) and m ≠ 0 (mod 3) and when m � 4, 5, and 7. We also investigate total number of total dominating sets for these cases.

Main Results
e following result provides a trivial lower bound on the total domination number of a graph in terms of the maximum degree of the graph.
is means that the set S � S 0 ∪ u 2 , u n−2 totally dominates all vertices in G.

Journal of Mathematics
For necessity, we need to prove the reverse inequality, i.e., c t (P(3k + 1, 4)) ≥ 2k + 4. e proof will be by mathematical induction. For n � 22, the result is obvious. Since the result is true for all P(n, 4) of the number of vertices less than n when m ≡ 0 (mod 3), for n ≡ 1 (mod 3), let D be any total dominating set of P(n, 4). en, D � D 1 ∪ u 2 ∪ u n−2 ∪ u n ∪ v n , where D 1 is the total dominating set of P(n, 4) if only n ≡ 0 (mod 3), and we get |D 1 | ≥ ⌈(2(n − 2))/3⌉ � 2k.
Assume the result is true for all P(n, 4) of the number of vertices less than n. For n ≡ 2 (mod3), let D be any total dominating set of P(n, 4). en, D � D 1 ∪ u 2 ∪ u n−2 ∪ u n−1 ∪ v n−1 , where D 1 is the total dominating set of P(n, 4) if only n ≡ 0 (mod3), and then |D 1 |≥⌈(2(n−2))/3⌉ � 2k, i.e., |D|≥2k+4. Proof. Let P 1 , P 2 , P 3 be the total domatic partitions since no vertex of any partition is contiguous to any other partitions and all the three domatic partitions totally dominate G (P(n, m)). Also, we know that TDV(u i ) � TDV(v i ) � 1. We proved in eorem 6 in the following that τ (P(n, m)) � 3 if only n ≡ 0 (mod 3). It is observed that there is no recurrence in the total number of total dominating sets. Hence, we can write d t (P(n, m)) � τ (P(n, m)) � 3 if only n ≡ 0 (mod 3). Henning and Yeo [24] proved the following.
(i) Subcase 1: when 〈Γ〉 ≡ kP 2 ∪ P 4 , note that Γ is fixed by the choice of single P 4 . Selecting P 4 is the same as selecting its initial vertex in the counterclockwise order; thus, τ(P(n, m)) � n. (ii) Subcase 2: when 〈Γ〉 ≡ (k − 1)P 2 ∪ 2P 3 , the set Γ is fixed by the choice of two P 3 's, and there are n options for choosing the first two P 3 's; thus, τ(P(n, m)) � n. (iii) Subcase 3: when 〈Γ〉 ≡ (k + 1)P 2 ∪ P 4 , there is only one chance for choosing single P 4 and (k + 1) chances for choosing P 2 . Each vertex has equal number of chances. Each vertex is equally repeated in the total number of total dominating sets three times; thus, the total number of total dominating sets will be n. (iv) Subcase 4: when 〈Γ〉 ≡ (k − 1)P 2 ∪ 2P 4 , then Γ is fixed by the choice of double P 4 . Each vertex is repeated three times in the total number of dominating sets, and for each vertex, there are equal number of chances for total domination; hence, each TDS will be repeated n times. (v) Subcase 5: when 〈Γ〉 ≡ (k + 1)P 2 , there are exactly two vertices in each P 2 . Each vertex of P 2 dominates three vertices including the vertex itself because this is a 3-regular graph. us, there are n number of choices for the total number of dominating sets; thus, in all cases, τ(P(n, m)) � n.
e proof is similar to the proof of eorem 3. □

Total Domatic Number of a Graph
e total domatic number of a graph G is denoted by d t (G) and is defined as d t (G) � s, where V(G) � ∪ s i�1 D i and D i ′ s are the total dominating sets and are the maximum disjoint partition of the vertex set of the graph.

Proposition 1. For any simple graph G with minimum degree δ(G), the total domatic number d t (G) ≤ δ(G).
In the following result, our proof is simple from the proof of [25].
Theorem 7 (see [25]). For a graph G of order n and minimum degree δ(G), the lower bound for the total domatic number Proof. Let D be the total dominating set of the graph G and G c be the complement of G. If δ(G) is the minimum degree of G, then n − δ(G) − 1 is the maximum degree of G c , and due to this, any vertex x of the graph G is adjacent to at most n − δ(G) − 1 vertices of D in G c . If the vertex x ∈ D, then a vertex z of the graph exists which is not adjacent to x in G c but xz ∈ E(G); this confirms that |D| ≥ n − δ(G) + 1, and hence, d t (G) ≥ ⌊n/(n − δ(G) + 1)⌋.

Open Problems
We conclude the paper with the following open problems for further investigation.
Open Problem 1. Characterize c t (P(n, m)) for all feasible values of n and m.

Discussion.
In this paper, the generalized Petersen graph (P(m, n)) is investigated for the total domination number c t , total number of dominating sets τ, and total domatic number d t (P(n, m)). Here, we found the total dominating number for some restricted values of m ∈ Z + . For detailed applications of our results, refer [26][27][28].

Data Availability
All data required for this paper are included within this paper.

Conflicts of Interest
e authors do not have any conflicts of interest.

Authors' Contributions
Taiyin Zhao gave applications of results and improved the presentation of the paper, Gohar Ali supervised this work, Nabila Hameed proved the results, Syed Inayat Ali Shah wrote the first version of the paper, and Yu-Ming Chu wrote the final version of the paper and arranged funding for this paper.