The connected and forcing connected restrained monophonic numbers of a graph

For a connected graph G = ( V, E ) of order at least two, a re-strained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by m r ( G ) . A connected restrained monophonic set S of G is a restrained mono-phonic set such that the subgraph G [ S ] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by m cr ( G ) . We determine bounds for it and ﬁ nd the same for some special classes of graphs. It is shown that, if a, b and p are positive integers such that 3 ≤ a ≤ b ≤ p, p − 1 6 = a, p − 1 6 = b , then there exists a connected graph G of order p with m r ( G ) = a and m cr ( G ) = b . Also, another parameter forcing connected restrained monophonic number f crm ( G ) of a graph G is introduced and several interesting results and realization theorems are proved.


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 [8].A block of a graph is a maximal non-separable subgraph.An end-block of G is a block containing exactly one cut-vertex of G.For vertices u and v in a connected graph G, the distance d(u, v) is the length of a shortest u − v path in G.An u − v path of length d(u, v) is called an u − v geodesic.It is known that d is a metric on the vertex set V of G.The neighborhood of a vertex v is the set N (v) consisting of all vertices u which are adjacent with v.The closed neighborhood of a vertex v is the set A vertex v is an extreme vertex if the subgraph induced by its neighbors is complete.The closed interval I[x, y] consists of all vertices lying on some x − y geodesic of G, while for S ⊆ V, I[S] = S x,y∈S I[x, y].A set S of vertices of G is a geodetic set if I[S] = V, and the minimum cardinality of a geodetic set is the geodetic number g(G).A geodetic set of cardinality g(G) is called a g-set.The geodetic number of a graph was introduced in [3,4,9] and further studied in [5,6].A geodetic set S of a graph G is a restrained geodetic set if the subgraph G[V − S] has no isolated vertex.The minimum cardinality of a restrained geodetic set of G is the restrained geodetic number.The restrained geodetic number of a graph was introduced and studied in [1].
A chord of a path P is an edge joining two non-adjacent vertices of P .A path P is called a monophonic path if it is a chordless path.A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G).The monophonic number of a graph and its related parameters was studied and discussed in [2,10,13,17,19].A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected.The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by m c (G).The connected monophonic number of a graph was introduced and studied in [7,20].A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices.The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by m r (G).The restrained monophonic number of a graph was introduced in [14] and further studied in [11,12,15,16,18].These concepts have interesting applications in Channel Assignment Problem in FM radio technologies.The monophonic matrix is used to discuss different aspects of certain molecular graphs associated to the molecules arising in special situations of molecular problems in theoretical chemistry.
The following theorems will be used in the sequel.
Theorem 1.1.[13] Each extreme vertex of a connected graph G belongs to every monophonic set of G.
Theorem 1.2.[20] Every cut-vertex of a connected graph G belongs to every connected monophonic set of G.
Theorem 1.4.[14] Let G be a connected graph with a cut-vertex v and let S be a restrained monophonic set of G. Then every component of G − v contains an element of S.
Theorem 1.5.[14] If T is a tree of order p with k end-vertices and p−k ≥ 2, then m r (T ) = k.
Theorem 1.6.[14] For the complete bipartite graph Throughout this paper G denotes a connected graph with at least two vertices.

Connected Restrained Monophonic Number
Definition 2.1.A connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected.The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by m cr (G).A connected restrained monophonic set of cardinality m cr (G) is called a m cr -set of G. Since every extreme vertex of a connected graph G belongs to every restrained monophonic set of G, we have the following observation.
Observation 2.3.Each extreme vertex of a connected graph G belongs to every connected restrained monophonic set of G.
From the Observation 2.3, it is clear that for the complete graph K p (p ≥ 2), m cr (K p ) = p.The next observation follows from Theorem 1.4.
Observation 2.4.Let G be a connected graph with cut-vertices and let S be a connected restrained monophonic set of G.If v is a cut-vertex of G, then every component of G − v contains an element of S.
Theorem 2.5.Every cut-vertex of a connected graph G belongs to every connected restrained monophonic set of G.

Proof.
Let v be any cut-vertex of G and let G 1 , G 2 , • • • , G r (r ≥ 2) be the components of G − v. Let S be any connected restrained monophonic set of G. Then by Observation 2.4, S contains at least one element from each Observation 2.6.For any connected graph G of order p with k extreme vertices and l cut-vertices, max{2, k + l} ≤ m cr (G) ≤ p.
For a cut-vertex v in a connected graph G and a component H of G − v, the subgraph H and the vertex v together with all edges joining v and V (H) is called a branch of G at v. Since every end-block B is a branch of G at some cut-vertex, it follows from Observation 2.4 that every minimum connected restrained monophonic set of G contains at least one vertex from B that is not a cut-vertex.Thus the following corollaries are consequences of Observation 2.4 and Theorem 2.5.
Corollary 2.9.For any non-trivial tree T of order p, m cr (T ) = p.
If p = 4, then it is clear that neither 2-element subset nor 3-element subset of V (C 4 ) forms a connected restrained monophonic set of C 4 and so m cr (C 4 ) = 4.
If p ≥ 5, then it is easily observed that any three consecutive vertices of G = C p forms a minimum connected restrained monophonic set of G = C p and so m cr (C p ) = 3.
2 In a complete bipartite graph, it is easy to observe that any minimum restrained monophonic set is also a connected restrained monophonic set.The next observation follows from Theorem 1.6.

Observation 2.11. For the complete bipartite graph
Proof.
Any restrained monophonic set needs at least two vertices and so m r (G) ≥ 2. Since every connected restrained monophonic set is also a restrained monophonic set, m r (G) ≤ m cr (G).Also, V (G) induces a connected restrained monophonic set of G. Hence, we have 2 ≤ m r (G) ≤ m cr (G) ≤ p. From the definition of the restrained monophonic number and the connected restrained monophonic number of a graph, we have 2 Observation 2.14.Let G be a connected graph of order p ≥ 2.
Proof.Any monophonic set needs at least two vertices and so m(G) ≥ 2. Since every connected monophonic set of G is a monophonic set of G, m(G) ≤ m c (G).Also, every connected restrained monophonic set of G is also a connected monophonic set of G and so  Proof.We prove this theorem by considering four cases.Since no vertex in V −S lies on a u−v monophonic path for some u, v ∈ S, S is not a monophonic set of G. Let S 1 = S ∪{v 1 }.It is easily verified that S 1 is a monophonic set of G and so m(G) = a.Let T = {v 3 , v 4 , • • • , v b−a+1 } be the set of all cut-vertices of G.By Theorems 1.1 and 1.2, every connected monophonic set of G contains S ∪ T .Let S 2 = S ∪ T .It is clear that S 2 is not a connected monophonic and connected restrained monophonic set of G. Observe that S 2 ∪ {v 1 } is a monophonic set of G and it is not a connected monophonic set of G, since the induced subgraph G[S 2 ] is not connected.Let S 3 = S 2 ∪ {v 1 , v 2 }.It is easily seen that S 3 is a connected monophonic set of G and so m c (G) = b.By Observation 2.3 and Theorem 2.5, every connected restrained monophonic set of G contains S ∪ T .Since the subgraph induced by V − S 3 has the isolated vertices w 1 , w 2 , . . ., w c−b .It is easy to observe that every connected restrained monophonic set of G contains all the vertices w 1 , w 2 , . . ., w c−b .Thus, S 3 ∪ {w 1 , w 2 , . . ., w c−b } is the unique minimum connected restrained monophonic set of G and so m cr (G) = c. 2

Forcing Connected Restrained Monophonic Number
For the graph G given in Figure 2.1, the sets S 3 = {u, v, w} and S 4 = {x, y, w} are the two minimum connected restrained monophonic sets of G and so m cr (G) = 3.Thus a connected graph G may contain more than one minimum connected restrained monophonic sets.For each minimum connected restrained monophonic set S in G, there is always some subset T of S that uniquely determines S as the minimum connected restrained monophonic set containing T, that is, T is not contained in any other minimum connected restrained monophonic set of G.Such sets are called "forcing connected restrained monophonic subsets" and we discuss these sets in this section.
Definition 3.1.Let S be a minimum connected restrained monophonic set of G.A subset T of a minimum connected restrained monophonic set S of G is called a forcing connected restrained monophonic subset for S if S is the unique minimum connected restrained monophonic set containing T .A forcing connected restrained monophonic subset for S of minimum cardinality is a minimum forcing connected restrained monophonic subset of S. The forcing connected restrained monophonic number of S, denoted by f crm (S), is the cardinality of a minimum forcing connected restrained monophonic subset for S. The forcing connected restrained monophonic number of G is f crm (G) = min{f crm (S)}, where the minimum is taken over all minimum connected restrained monophonic sets S in G.The next theorem follows immediately from the definitions of the connected restrained monophonic number and the forcing connected restrained monophonic number of a graph G.
The following theorem is an easy consequence of the definitions of the connected restrained monophonic number and forcing connected restrained monophonic number.In fact, the theorem characterizes graphs G for which the lower bound in Theorem 3.3 is attained and also graphs G for which f crm (G) = 1 and f crm (G) = m cr (G).Theorem 3.4.Let G be a connected graph.Then (i) f crm (G) = 0 if and only if G has the unique minimum connected restrained monophonic set.(ii) f crm (G) = 1 if and only if G has at least two minimum connected restrained monophonic sets, one of which is a unique minimum connected restrained monophonic set containing one of its elements, and (iii) f crm (G) = m cr (G) if and only if no minimum connected restrained monophonic set of G is the unique minimum connected restrained monophonic set containing any of its proper subsets.
A vertex v of a connected graph G is said to be a connected restrained monophonic vertex of G if v belongs to every minimum connected restrained monophonic set of G. Observation 3.5.If G has an unique minimum connected restrained monophonic set S, then every vertex in S is a connected restrained monophonic vertex of G. Also, if x is an extreme vertex or a cut-vertex of G, then by Observation 2.3 and Theorem 2.5, x is a connected restrained monophonic vertex of G.
The following theorem and corollary follow immediately from the definitions of connected restrained monophonic vertex and forcing connected restrained monophonic subset of G. Theorem 3.6.Let G be a connected graph and let ρ dm be the set of relative complements of the minimum forcing connected restrained monophonic subsets in their respective minimum connected restrained monophonic sets in G. Then T F ∈ρ dm F is the set of connected restrained monophonic vertices of G. Corollary 3.7.Let G be a connected graph and let S be a minimum connected restrained monophonic set of G. Then no connected restrained monophonic vertex of G belongs to any minimum forcing connected restrained monophonic subset of S.
Let S be any minimum connected restrained monophonic set of G. Then

Proof.
Let m cr (G) = 2. Then by Observation 2.14, S = V (G) is the unique minimum connected restrained monophonic set of G.It follows from Theorem 3.4(i) that f crm (G) = 0.
2 Now, we proceed to determine the forcing connected restrained monophonic number of certain classes of graphs.The next observation follows from Theorem 2.10.

Proof.
For G = K p , it follows from Observation 2.3, that the set of all vertices of G is the unique minimum connected restrained monophonic set of G. Now, it follows from Theorem 3.4(i) that f crm (G) = 0.If G is a nontrivial tree, then by Corollary 2.9, the set of all vertices of G is the unique minimum connected restrained monophonic set of G and so by Theorem

Figure 2 .
Figure 2.2: Graph G Remark 2.13.The bounds for the Theorem 2.12 are sharp.If G = K p , m r (G) = p and m cr (G) = p.All the inequalities in Theorem 2.12 can be strict.For the graph G given in Figure 2.2, S = {v 1 , v 5 , v 6 } is the unique minimum restrained monophonic set of G so that m r (G) = 3, and no 3element or no 4-element subset of the vertex set is a connected restrained monophonic set of G. Since {v 1 , v 4 , v 5 , v 6 , v 7 } is a connected restrained monophonic set of G, it follows that m cr (G) = 5.Thus, we have 2 < m r (G) < m cr (G) < p.

2 Remark 2 . 16 .
From the definition of the connected restrained monophonic number of a graph, we have m cr (G) 6 = p − 1.The bounds for the Theorem 2.15 are sharp.If G = K p , then m(G) = m c (G) = p and m cr (G) = p.All the inequalities in Theorem 2.15 can be strict.For the graph G given in Figure 2.3, S 1 = {v 1 , v 3 , v 5 } is the unique minimum monophonic set of G so that m(G) = 3 and S 2 = {v 1 , v 2 , v 3 , v 5 } is a connected monophonic set of G and so m c (G) = 4.It is easily verified that S 2 = {v 1 , v 2 , v 3 , v 5 } is not a connected restrained monophonic set of G since the subgraph induced by V − S 2 has an isolated vertex v 4 .Hence S 2 ∪{v 4 } is a connected restrained monophonic set of G.It follows that m cr (G) = 5.Thus, we have 2 < m(G) < m c (G) < m cr (G) < p.

Figure 2 . 3 : 2 Remark 2 . 18 .Problem 2 . 19 .
Figure 2.3: Graph G Theorem 2.17.Let G be a connected graph of order p with every vertex of G either a cut-vertex or an extreme vertex.Then m cr (G) = p.

Case 1 .
3 ≤ a = b = p.Let G be a complete graph of order p.Then by Theorem 1.3 and Observation 2.3, G has the desired properties.

Figure 3 . 1 : Graph G Example 3 . 2 .
Figure 3.1: Graph G Example 3.2.For the graph G given in Figure 2.1, S 3 = {u, v, w} and S 4 = {x, y, w} are the two minimum connected restrained monophonic sets of G.It is clear that f crm (S 3 ) = 1 and f crm (S 4 ) = 1 so that f crm (G) = 1.For the graph G given in Figure 3.1, S = {v 1 , v 2 , v 3 , v 4 } is the unique minimum connected restrained monophonic set of G and so f crm (G) = 0.

Theorem 3 . 3 .
For a connected graph G of order p, 0 ≤ f crm (G) ≤ m cr (G) ≤ p, p − 1 6 = m cr (G).The bounds in Theorem 3.3 are sharp.For the graph G given in Figure3.1, f crm (G) = 0.By Observation 2.3, for the complete graph K p (p ≥ 2), m cr (K p ) = p.The inequalities in Theorem 3.3 can be strict.For the graph G given in Figure2.1, m cr (G) = 3 and f crm

m
cr (G) = |S|, M ⊆ S and S is the unique minimum connected restrained monophonic set containing S − M .Thus f crm (G) ≤ |S − M | = |S| − |M | = m cr (G) − |M |.Theorem 3.8.Let G be a connected graph and let M be the set of all connected restrained monophonic vertices of G. Then f crm (G) ≤ m cr (G) − |M |.Corollary 3.9.If G is a connected graph with l extreme vertices and k cut-vertices, then f crm (G) ≤ m cr (G) − (l + k).The bound in Theorem 3.8 is sharp.For the graph G given in Figure 3.1, m cr (G) = 4 and f crm (G) = 0. Also, M = {v 1 , v 2 , v 3 , v 4 } is the unique minimum connected restrained monophonic set of G.By Observation 3.5, every vertex of M is a connected restrained monophonic vertex of G and so f crm (G) = m cr (G) − |M |.Also the inequality in Theorem 3.8 can be strict.For the graph G given in Figure 2.1, S 3 = {u, v, w} and S 4 = {x, y, w} are the minimum connected restrained monophonic sets so that m cr (G) = 3 and f crm (G) = 1.Also, since only one vertex w of G is a connected restrained monophonic vertex of G, we have f crm (G) < m cr (G) − |M |.Theorem 3.10.If G is a connected graph with m cr (G) = 2, then f crm (G) = 0.

2 Theorem 3 . 13 .
3.4(i), f crm (G) = 0.For the complete bipartite graphG = K m,n (2 ≤ m ≤ n), f crm (G) = ( 0 if 2 = m ≤ n 4 if 3 ≤ m ≤ n.Proof.Let U = {x 1 , x 2 , . . ., x m } and W = {y 1 , y 2 , . . ., y n } be the partite sets of G, where m ≤ n.We prove this theorem by considering two cases.Case 1. 2 = m ≤ n.By Observation 2.11, V (G) is the unique minimum connected restrained monophonic set of G and so by Theorem 3.4(i),f crm (G) = 0. Case 3. 3 = m ≤ n.Then any minimum connected restrained monophonic set is got by choosing any two elements from each of U and W, and G has at least two minimum connected restrained monophonic sets.Hence m cr (G) = 4. Clearly, no minimum connected restrained monophonic set of G is the unique minimum connected restrained monophonic set containing any of its proper subsets.Then by Theorem 3.4(iii), we have f crm (G) = m cr (G) = 4.2