The forcing monophonic and the forcing geodetic numbers of a graph

For a connected graph G = (V,E), let a set S be a m-set of G. A subset T ⊆ S is called a forcing subset for S if S is the unique m-set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing monophonic number of S, denoted by fm(S), is the cardinality of a minimum forcing subset of S. The forcing monophonic number of G, denoted by fm(G), is fm(G) = min{fm(S)}, where the minimum is taken over all minimum monophonic sets in G. We know that m(G) ≤ g(G), where m(G) and g(G) are monophonic number and geodetic number of a connected graph G respectively. However there is no relationship between fm(G) and fg(G), where fg(G) is the forcing geodetic number of a connected graph G. We give a series of realization results for various possibilities of these four parameters.


Introduction
By a graph G = (V,E), we mean a finite undirected connected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. For basic graph theoretic terminology, we refer to Harary [1]. The distance d(u, v) between two vertices u and v in a connected graph G is the length of shortest u − v path in G. An u − v path of length d(u, v) is called an u − v geodesic. A vertex x is said to be lie a u − v geodesic P if x is a vertex of P including the vertices u and v. A geodetic set of G is a set S ⊆ V such that every vertex of G is contained in geodesic joining some pair of vertices in S. The geodetic number g(G) of G is the minimum order of its geodetic sets and any geodetic set of order g(G) is a minimum geodetic set or simply a g-set of G. The geodetic number of a graph was introduced in [1] and further studied in [3,4,5,7,8,9,16,17,18,20,23,25]. A subset T ⊆ S is called a forcing subset for S if S is the unique g-set of G containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing geodetic number of S, denoted by f g (S), is the cardinality of a minimum forcing subset of S. The forcing geodetic number of G, denoted by f g (G), is f g (G) = min{f g (S)}, where the minimum is taken over all minimum g-sets of G. The forcing geodetic number of a graph was introduced in [3] and furthur studied in [19,21,22]. A chord of the path P is an edge joining to non-adjacent vertices of P . An u − v path P is called monophonic path if it is a chordless path. A monophonic set of G is a set S ⊆ V such that every vertex of G is contained in a monophonic path joining some pair of vertices in M. The monophonic number m(G) of G is the minimum order of its monophonic sets and any monophonic set of order m(G) is a minimum monophonic set or simply a m-set of G. The monophonic number of a graph was introduced in [6] and further studied in [2,6,10,11,12,13,14,15,19,24]. A vertex v is said to be monophonic vertex of G if v belongs to every minimum monophonic set of G. A vertex v is an extreme vertex of a graph G if the sub graph induced by its neighbours is complete. A vertex v is said to be geodetic(monophonic) vertex if v belongs to every g-set (m-set) of G. Every extreme vertices are geodetic(monophonic) vertices of G. In fact there are geodetic (monophonic) vertices which are not extreme vertices of G. Let G be a connected graph and S a m-set of G. A subset T ⊆ S is called a forcing subset for S if S is the unique m-set of G containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The f orcing monophonic number of S, denoted by f m (S), is the cardinality of a minimum forcing subset of S. The forcing monophonic number of G, denoted by f m (G) is defined by f m (G) = min {f m (S)}, where the minimum is taken over all m-sets S in G. The forcing monophonic number of a graph was introduced in [11]. The Throughout the following G denotes a connected graph with at least two vertices.The following theorems are used in the sequel. Theorem 1.1. [4,12] If v is an extreme vertex of a connected graph G, then v belongs to every geodetic (monophonic) set of G.

The Forcing Monophonic and the Forcing Geodetic Numbers of a Graph
We know that m(G) ≤ g(G). From the following examples, we observe that there is no relationship between f m (G) and f g (G).

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The forcing monophonic and the forcing geodetic numbers of a graph | J. John Example 2.1. For the graph G given in Figure 2 Example 2.2. For the graph G given in Figure 2

Special graphs
In this section, we present some graphs from which various graphs arising in theorem are generated using identification.
Let P i : u i , v i be a copy of paths on two vertices. Let G a be the graph given in Figure 3.1 obtained from P i (≤ i ≤ a) by introducing new vertices s, t and joining each u i (1 ≤ i ≤ a) with s and joining each v i (1 ≤ i ≤ a) with t and join s with t.

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The forcing monophonic and the forcing geodetic numbers of a graph | J. John be a copy of path on two vertices and P : l, m, n be a path on three vertices. Let Z b be the graph given in Figure 3 be a copy of path on three vertices and let P : e, f, g be a path on three vertices. Let H c be the graph given in Figure 3.3 obtained from P i (1 ≤ i ≤ c) and P by joining e and f with each h i and r The forcing monophonic and the forcing geodetic numbers of a graph | J. John be the path on three vertices. Let R a be the graph given in Figure  3.4 obtained from U i (1 ≤ i ≤ d) by adding new vertices u and v by joining u with v and joining each does not lie on any geodesic joining a pair of vertices in Z, we see that Z is not a geodetic set of G. It is easily verified that We observed that every g-set of G must contain at least one vertex from each = a. Now, since g(G) = c and every g-set of G contains W 1 , it is easily seen that every g-set S is of the form Let T be any proper subset of S with | T |< a. Then it is clear that there exists some j such that T ∩ Q j = Φ, which shows that f g (G) = a.
We observe that every g-set of G must contain only the vertex y i from each H i (1 ≤ i ≤ a) and so g(G) ≥ b − a + a = b. Now S = Z ∪ {y 1 , y 2 , y 3 , . . . , y a } is a geodetic set of G so that g(G) ≤ b − a + a = b. Thus g(G) = b. Also it is easily seen that W is the unique g-set of G and so f g (G) = 0. Now it is clear that Z is not a monophonic set of G. We observe that every m-set of G must contain at least one vertex from each H i (1 ≤ i ≤ a). Hence by Theorem 1.1, m(G) ≥ b − a + a = b. Now We observe that every g-set of G must contain only the vertex Also it is easily seen that W is the unique g-set of G and so f g (G) = 0. It is clear that Z is not a monophonic set of G. We observe that every m-set of G must contain at least one vertex from each  Since the vertices u i , v i do not lie on any monophonic path joining a pair of vertices of Z, it is clear that Z is not a monophonic set of G.
We observe that every m-set of G must contain at least one vertex from each F i (1 ≤ i ≤ a).
Let T be any proper subset of S with | T |< a. Then it is clear that there exists some j such that T ∩ H j = Φ, which shows that f m (G) = a. Next we show that g(G) = c. Since the vertices do not lie on any geodesic joining a pair of vertices of Z, it is clear that Z is not a geodetic set of G. We observe that every g-set of G must contain each . . , u a } is a geodetic set of G, so that g(G) ≤| S 1 |= c. Hence g(G) = c. Next we show that f g (G) = a. By Theorem 1.1, every geodetic set of G contains W 1 = Z ∪ {h 1 , h 2 , h 3 , . . . , h c−b } and so it follows from Theorem 1.3(b) that f g (G) ≤ g(G)− | W 1 |= a. Now, since g(G) = c and every g-set of G contains Z, it is easily seen that every g-set S is of the form Let T be any proper subset of S with | T |< a. Then it is clear that there exists some j such that T ∩ H j = Φ, which shows that f g (G) = a. This is true for all g-sets of G so that f g (G) = a.

The Upper Forcing Monophonic number of a graph
In [25], P. Zhang introduced the concept of the upper geodetic number of a graph. In the similar manner we define the upper forcing monophonic number of a graph as follows.