StrongTotalMonophonicProblems inProductGraphs,Networks, and Its Computational Complexity

Department of Mathematics, Loyola College (A iated to the University of Madras), Chennai, India Department of Studies in Mathematics, University of Mysore, Manasagangothri, Mysuru 570 006, Karnataka, India Department of Mathematics, St. Joseph’s College, Bangalore, India Department of Mathematics, St. Joseph’s College of Engineering, Chennai, India Department of Mathematics, Ibb University, Ibb, Yemen

is concept is introduced by Harary et al. [3]. If S⊆V, then for each pair of vertices x, y ∈ S, x ≠ y, let g(x, y) be a selected xed shortest x − y path. en, we set I(S) g(x, y): x, y ∈ S }, and let V( I(S)) ∪ P∈ I(S) V(P). If V( I(S)) V, for some I(S), then the set S is called a strong geodetic set, and the strong geodetic problem is to nd a minimum strong geodetic set S of G [5].
A p − q monophonic path is a chordless p − q path. e monophonic distance is the number of edges of the chordless path, denoted by d m (p, q). e monophonic eccentricity of a vertex p ∈ V(G) is the maximum monophonic distance between p and another vertex in G. e monophonic diameter, d m , and monophonic radius, r m , of G are the maximum and minimum monophonic eccentricity among the vertices in G, respectively. A set of vertices S⊆Vis de ned as monophonic set (MS) if every vertex of G\S lies on a monophonic path joining some pair of vertices in S, and the cardinality of such smallest set is called monophonic number, denoted by m(G). We refer [6][7][8][9][10] for other parameters of monophonic number. If every vertex of G\S lies on a unique xed monophonic path between the pair of vertices in S⊆V(G), then S is said to be a strong monophonic set (SMS, for short) which is de ned in [11]. Its minimum cardinality is called strong monophonic number, denoted by sm(G).

Motivation
Graph theory plays a vital role in modelling our day to day activities into graphs. We can model the traffic system in a region using graphs where the vertices are junctions of roads. e police officers can be stationed in such a way that (i) two police officers are stationed at junctions adjacent so that they can support each other if they face any issue, and (ii) two police officers other than the former ones are stationed in such a way that they guard the junctions along the unique path than the former pair of police officers. us, we can assign minimum number of police officers as well as reduce the risk of traffic issues. For the above mentioned problem, the concept of strong total monophonic problem is helpful.
is concept can also be useful in power supply station, water supply station, etc. e next section defines the problem, and its computational complexity and results are discussed in the following sections.

Strong Total Monophonic Number of a Graph
A total monophonic set (TMS, for short), S⊆V(G), is a monophonic set whose induced subgraph does not contain any isolated vertices. e cardinality of minimum total monophonic set is called total monophonic number (TM, for short), denoted by m t (G). A set T⊆V(G) is said to be a strong total monophonic set (STMS, for short) if T is a strong monophonic set, and the subgraph induced by T does not contain isolated vertices. e cardinality of such smallest STMS is called strong total monophonic number, sm t (G).
In Figure 1, the TMS is a, b, f, g and m t (S) � 4. e STMS is a, b, c, d, e { } and sm t (S) � 5. We recall some terms and definitions required. Simplicial vertices of a graph G are those vertices whose neighbourhood induces a clique and is denoted by Ext(G). e vertex adjacent to the vertex of degree 1, that is, end vertex is called stem vertex. e sets of all end vertices and all stems are denoted by Ω(G) and Ω 1 (G), respectively [12].

Proposition 1.
Every Ω(G) and Ω 1 (G) of a connected graph G belongs to every STMS of G. If the set of all Ω(G) and Ω 1 (G) forms a STMS, then it is the unique minimum strong total monophonic set of G.
e set of all Ω(T) and Ω 1 (T) is the unique minimum STMS for any nontrivial tree T.
Proof. Since extreme and stem vertices of a nontrivial tree T form a STMS of T, the result is obtained from Proposition 1.

Proof
(i) A strong monophonic set, SMS, contains a minimum of 2 vertices or more; thus, sm(H) ≥ 2. Each STMS is a strong monophonic set; therefore,

Complexity Results
We are reducing the problem of strong total monophonic for general graphs from the decision problem, whether there exists an induced path between two vertices such that it passes through a third vertex. Using the above concept, we solve NP-completeness of STMS.
Theorem 3 (See [13]). Let x, y, z be three distinct vertices in a graph G. Deciding whether there is an induced path from x to y passing through z is NP-complete.

Theorem 4.
e STM problem for general graphs is NPcomplete.
Proof. Given a graph H with distinct vertices p, q, r and construct the graph H ′ as follows: let p ′ , q ′ ∈ V(H ′ ) such that these vertices are adjacent to all the vertices in V(H) r { }. A pendant vertex a is joined to the vertex p ′ . Similarly, pendant vertices a 1 , a 2 , . . . , a n− 1 are joined to the vertex v, refer Figure 2. Let S � p ′ , q ′ , a, a 1 , a 2 , . . . , a n− 1 since the set of vertices of T forms a set of Ext(H) and stem vertices, which belong to any STMS of H ′ . Also, the monophonic paths between the vertices in T do not cover the vertex r. It is straightforward to see that every induced path in H is also an induced path in H ′ . e monophonic paths between x and x ′ where x ′ ∈ S and x ∈ V(H) will not cover the vertex r.

Theorem 5. Let H be a graph of order
□ e above bound is sharp for Figure 3.
Proof. If H � K 2 , then sm t (H) � 2. Conversely, assume S � x, y be a minimum STMS of H. Suppose the length of x − y monophonic path is greater than 2, then it has an internal vertex. Let it be w, such that w ∈ V(H)\S. erefore, G[S] has isolated vertices, which is a contradiction. erefore, the only possibility is H � K 2 . Proof. Clearly, for path P 3 , sm t (H) � 3.
Let H be cycle C n : v 1 , v 2 , . . . , v n of order n. By eorem e set S is a strong monophonic set of C n . Hence, sm(C n ) � 3.
Conversely, let sm t (H) � 3. Let us assume that H is not a path P 3 Sum of two graphs S and T is a graph with vertices and

Theorem 9. Let S and T be two connected graphs. Let
is not applicable to a strong total monophonic number of G + H. Consider P 3 + P 4 , as in Figure 4, the total monophonic set S contains a, c, d { } where a, c { } is the monophonic set of P 3 and S ∩ P 4 � d { } whereas the strong total monophonic set is a, c, d, g .

Theorem 10. Let S and T be two noncomplete graphs, then
p r q p q a a 1 a 2 a 3 a n-1 p' q' r  Proof. Let Z and Z ′ be the SMS of S and T, respectively.
□ e bound is sharp for P n + C m . e corona product of two graphs S and T, S°T, is the graph obtained by joining every vertex of k th copy of T to k th vertex of S. Theorem 12. Let X and X ′ be two connected graphs, then sm t (X°X ′ ) ≤ n(X)(1 + sm(X ′ )).
□ e sharpness for the above bound is obtained for P n°Pm , n, m ≥ 3.
Cartesian product of two graphs S and T, S□T is a graph with vertex set V(S) × V(T), and two vertices (s, t) and (s ′ , t ′ ) are adjacent in S□T if and only if s � s ′ and t is adjacent to t ′ or t � t ′ and s is adjacent to s ′ . Proof. By eorem 13, sm(S□T) ≤ mn − 1. By eorem 2, sm t (S□T) ≤ 2(mn − 1).

□ Theorem 15. Let S and T be two connected graphs. en, sm t (S□T) ≤ min sm(S)sm t (T), sm t (S)sm(T) .
en, there exists a fixed monophonic path P: (l i , l j ) where l i , l j ∈ X such that x lies on P i path. Similarly, y lies on a fixed monophonic path Q: (l i ′ , l j ′ ) where l i ′ , l j ′ ∈ Y. By Remark 1, (x, y) will lie on some fixed monophonic path with adjacent edges P and Q and corner vertices (l i , l i ′ ), (l i , l j ′ ), (l j , l i ′ ), (l j , l j ′ ). Since Y is a strong total monophonic set, G[X × Y] contains no isolated vertices. Hence, sm t (S□T) ≤ ll ′ . By symmetry, sm t (S□T) ≤ sm t (S)sm(T).

erefore, sm t (S□T) ≤ min sm(S)sm t (T), sm t (S)sm(T) .
□ e bound is sharp for P n □P 2 . Strong product of two graphs S and T, S⊠T, is a graph with vertex set V(S) × V(T), and two vertices (s, t) and (s ′ , t ′ ) are adjacent in S⊠T if and only if s � s ′ and t is adjacent to t ′ or t � t ′ and s is adjacent to s ′ or t is adjacent to t ′ and s is adjacent to s ′ .

Theorem 16. For two connected graphs S and T, sm t (S⊠T) ≤ min sm(S)sm t (T), sm t (S)sm(T) .
e proof is similar to eorem 15. Sharpness of the bound is obtained for P n ⊠P 2 .
Proof. Let the vertices of P n ⊠P m be (a i , b j ): 1 ≤ i ≤ m; 1 ≤ j ≤ n}. By eorem 7, sm t (G) ≥ 4. Denote S � (a 1 , b 1 ), (a 1 , b n ), (a m , b 1 ), (a m , b n )}. All the internal and external vertices are covered by a unique fixed chordless path between the pair of vertices in S. us, all the vertices in P n ⊠P m lie on some unique fixed monophonic path. Since G[S] contains isolated vertices, at most one vertex adjacent to each vertex in the set S is also considered along with S to form the strong total monophonic number. us, sm t (P n ⊠P m ) ≤ 8. Proof. By Corollary 2, sm t (G) ≥ 4. Consider the set S⊆V(G) such that S � x, y, z, w and x is adjacent to y and z is adjacent to w. e vertices are covered by unique fixed x − w, z − w, and z − y monophonic paths as given in Figure 5. us, S is a STMS, and sm t (G) � 4.

Theorem 19.
e strong total monophonic number of the Sirpenski graph S(n, k) of dimension, n ≥ 2, is 6. Proof. In the Sierpinski graph, the corner vertices form simplicial vertices. Also, Ext(S(n, k)) ∈ STMS. e Ext (S(n, k)) covers all the vertices in a unique fixed monophonic path. us, |Ext(S(n, k))| � 3. By definition of a strong total monophonic number, the set must consist of no isolated vertices; therefore, the strong total monophonic number of the Sierpinski graph is 6, that is, twice the number of simplicial vertices. □ Theorem 20. For n-dimensional silicate network SL(n), sm t (SL(n)) � 6n 2 + 6n + (n(n − 1)/2).
Proof. By Corollary 2, sm t (W n ) ≥ 4. Let W n � C n− 1 + K 1 be the wheel graph of order n, n ≥ 5 which is constructed as follows. Let u 0 be center vertex which is adjacent to all vertices of C n− 1 , n ≥ 5 and the diameter of wheel graph is 2, for all n ≥ 5. e vertices that lie on the circle are denoted by u 1 , u 2 , . . . , u n− 1 .
e u 1 − u n− 2 unique fixed monophonic path covers all vertices except u 0 , u n− 1 .

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.