МАТЕМАТИКА Forcing total outer connected monophonic number of a graph

. For a connected graph 𝐺 = ( 𝑉, 𝐸 ) of order at least two, a subset 𝑇 of a minimum total outer connected monophonic set 𝑆 of 𝐺 is a forcing total outer connected monophonic subset for 𝑆 if 𝑆 is the unique minimum total outer connected monophonic set containing 𝑇 . A forcing total outer connected monophonic subset for 𝑆 of minimum cardinality is a minimum forcing total outer connected monophonic subset of 𝑆 . The forcing total outer connected monophonic number 𝑓 𝑡𝑜𝑚 ( 𝑆 ) in 𝐺 is the cardinality of a minimum forcing total outer connected monophonic subset of 𝑆 . The forcing total outer connected monophonic number of 𝐺 is 𝑓 𝑡𝑜𝑚 ( 𝐺 ) = min { 𝑓 𝑡𝑜𝑚 ( 𝑆 ) } , where the minimum is taken over all minimum total outer connected monophonic sets 𝑆 in 𝐺 . We determine bounds for it and find the forcing total outer connected monophonic number of a certain class of graphs. It is shown that for every pair 𝑎, 𝑏 of positive integers with 0 (cid:54) 𝑎 < 𝑏 and 𝑏 (cid:62) 𝑎 + 4 , there exists a connected graph 𝐺 such that 𝑓 𝑡𝑜𝑚 ( 𝐺 ) = 𝑎 and 𝑐𝑚 𝑡𝑜 ( 𝐺 ) = 𝑏 , where 𝑐𝑚 𝑡𝑜 ( 𝐺 ) is the total outer connected monophonic number of a graph.


Introduction
By a graph = ( , ) we mean a finite simple undirected connected graph. The order and size of are denoted by and , respectively. For basic graph theoretic terminology we refer to Harary [1,2]. The distance ( , ) between two vertices and in a connected graph is the length of a shortest − path in . An − path of length ( , ) is called an − geodesic. A vertex of a connected graph is called an endvertex of if its degree is 1. A vertex of a connected graph is called a support vertex of if it is adjacent to an endvertex of . The ℎ ℎ of a vertex is the set ( ) consisting of all vertices which are adjacent with . A vertex is an extreme vertex if the subgraph induced by its neighbors is complete. A ℎ of a path is an edge joining two non-adjacent vertices of . A path is called a monophonic path if it is a chordless path. A set of vertices of is a monophonic set of if each vertex of lies on a − monophonic path for some and in . The minimum cardinality of a monophonic set of is the monophonic number of and is denoted by ( ). The monophonic number of a graph, an algorithmic aspect of monophonic concepts was introduced and studied in [3][4][5][6][7]. A total monophonic set of a graph is a monophonic set such that the subgraph [ ] induced by has no isolated vertices. The minimum cardinality of a total monophonic set of is the total monophonic number of and is denoted by ( ). The total monophonic number of a graph and its related concepts were studied in [8][9][10]. A set of vertices in a graph is said to be an outer connected monophonic set if is a monophonic set of and either = or the subgraph induced by − is connected. The minimum cardinality of an outer connected monophonic set of is the outer connected monophonic number of and is denoted by ( ). The outer connected monophonic number of a graph was introduced in [11]. Very recently, outer connected monophonic concepts have been widely investigated in graph theory, such as a connected outer connected monophonic number [12], extreme outer connected monophonic graphs [13], and so on. A total outer connected monophonic set of is an outer connected monophonic set such that the subgraph induced by has no isolated vertices. The minimum cardinality of a total outer connected monophonic set of is the total outer connected monophonic number of and is denoted by ( ). The authors of this article introduced and studied the general externally total outer connected monophonic number of a graph and proved the following theorems 1 , which will be used further.  Throughout this paper, denotes a connected graph with at least two vertices.

Main Results
Definition 1. Let be a minimum total outer connected monophonic set of . A subset of is a forcing total outer connected monophonic subset for if is the unique minimum total outer connected monophonic set containing . A forcing total outer connected monophonic subset for of minimum cardinality is a minimum forcing total outer connected monophonic subset of . The forcing total outer connected monophonic number ( ) in is the cardinality of a minimum forcing total outer connected monophonic subset of . The forcing total outer connected monophonic number of is ( ) = min{ ( )}, where the minimum is taken over all minimum total outer connected monophonic sets in . Example 1. For the graph in Fig. 1, it is clear that } are the minimum total outer connected monophonic sets of . It is clear that no minimum total outer connected monophonic set ( = 1, 2, 3, 4) is the unique minimum total outer connected monophonic set containing any of its 1-element subsets. It is easy to see that { 2 , 4 } is a forcing total outer connected monophonic subset contained in 1 and ( 1 ) = 2. Hence, we have ( ) = 2. By Theorem 3, for any non-trivial tree , the set of all endvertices and support vertices of is the unique minimum total outer connected monophonic set of and so ( ) = 0. The following theorem characterizes graphs for which the lower bound in Theorem 5 is attained and also characterizes graphs for which ( ) = 1 and ( ) = ( ).

Theorem 6. Let be a connected graph. Then (i) ( ) = 0 if and only if has the unique minimum total outer connected monophonic set;
(ii) ( ) = 1 if and only if has at least two minimum total outer connected monophonic sets, one of which is the unique minimum total outer connected monophonic set containing one of its elements; (iii) ( ) = ( ) if and only if no minimum total outer connected monophonic set of is the unique minimum total outer connected monophonic set containing any of its proper subsets.

Proof. (i) Let
( ) = 0. Then, by the definition, ( ) = 0 for some minimum total outer connected monophonic set of so that the empty set is the minimum forcing subset for . Since the empty set is a subset of every set, it follows that is the unique minimum total outer connected monophonic set of . The converse is clear.
(ii) Let ( ) = 1. Then by (i), has at least two minimum total outer connected monophonic sets. Since ( ) = 1, there is a 1-element subset of a minimum total outer connected monophonic set of such that is not a subset of any other minimum total outer connected monophonic set of . Thus is the unique minimum total outer connected monophonic set containing one of its elements. The converse is clear.
(iii) Let ( ) = ( ). Then ( ) = ( ) for every minimum total outer connected monophonic set in . Since any total outer connected monophonic set of needs at least two vertices, ( ) 2 and hence ( ) 2. Then by (i), has at least two minimum total outer connected monophonic sets, and so the empty set is not a forcing subset for any minimum total outer connected monophonic set of . Since ( ) = ( ), no proper subset of is a forcing subset of . Thus no minimum total outer connected monophonic set of is the unique minimum total outer connected monophonic set containing any of its proper subsets.
Conversely, the data implies that contains more than one minimum total outer connected monophonic set, and no subset of any minimum total outer connected monophonic set other than , is a forcing subset for . Hence it follows that ( ) = ( ).

Definition 2.
A vertex of is said to be a total outer connected monophonic vertex if belongs to every minimum total outer connected monophonic set of .

Remark 2.
If has the unique minimum total outer connected monophonic set , then every vertex in is a total outer connected monophonic vertex of . Also, if is an extreme vertex or a support vertex of , then is a total outer connected monophonic vertex of . For the graph given in Fig. 1, 1 and 5 are the total outer connected monophonic vertices of .
The next theorem and corollary are an immediate consequence of the definitions of total outer connected monophonic vertex and a forcing total outer connected monophonic subset of .

Theorem 7.
Let be a connected graph and let Ψ be the set of relative complements of the minimum forcing total outer connected monophonic subsets in their respective minimum total outer connected monophonic sets in . Then ∩ ∈Ψ is the set of all total outer connected monophonic vertices of . Remark. 3. The bound in Theorem 8 is sharp. For the graph given in Fig. 1, the minimum total outer connected monophonic sets of It is clear that ( ) = 2( = 1, 2, 3, 4) and so ( ) = 2. Also, It is easy to verify that, no minimum total outer connected monophonic set of is the unique minimum total outer connected monophonic set containing any of its proper subsets. Then by Theorem 6 (iii), we have ( ) = + − 1. Case 2. If 3 , then any minimum total outer connected monophonic set of is obtained by choosing any two elements from as well as , and has at least two minimum total outer connected monophonic sets. Hence ( ) = 4. Clearly, no minimum total outer connected monophonic set of is the unique minimum total outer connected monophonic set containing any of its proper subsets. Then by Theorem 6 (iii), we have ( ) = ( ) = 4.

Theorem 11. For any cycle
Proof. Let : 1 , 2 , . . . , , 1 be a cycle of order . We prove this theorem by considering two cases.
Case 1: = 3. Since 3 is the complete graph of order 3, ( 3 ) is the unique minimum total outer connected monophonic set of 3 . By Theorem 6 (i), ( 3 ) = 0. Case 2: 4. It is clear that no 2-element subset of ( ) is a total outer connected monophonic set of . It is easy to verify that any minimum total outer connected monophonic set of consists of three consecutive vertices of so that ( ) = 3. For = 4, it is clear that no minimum total outer connected monophonic set of 4 is the unique minimum total outer connected monophonic set containing any of its proper subsets. Thus by Theorem 6 (iii), we have ( 4 ) = 3. For 5, it is clear that the set of two non-adjacent vertices of any minimum total outer connected monophonic set of is a minimum forcing total outer connected monophonic subset of and so ( ) = 2. Hence ( ) = 2. Proof. It is clear that no 2-element subset of ( ) is a total outer connected monophonic set of . It is easy to observe that any minimum total outer connected monophonic set of consists of three consecutive vertices of −1 so that ( ) = 3. For = 5, it is clear that no minimum total outer connected monophonic set of 5 is the unique minimum total outer connected monophonic set containing any of its proper subsets. Thus by Theorem 6 (iii), we have ( 5 ) = 3. For 6, it is clear that the set of two non-adjacent vertices of any minimum total outer connected monophonic set of is a minimum forcing total outer connected monophonic subset of and so ( ) = 2. Hence ( ) = 2.
Proof. Let = . By Theorem 2, the set of all vertices of is the unique minimum total outer connected monophonic set of and so by Theorem 6 (i), ( ) = 0. If is a non-trivial tree, then by Theorem 3, the set of all endvertices and support vertices of is the unique minimum total outer connected monophonic set of and by Theorem 6 (i), ( ) = 0.
Theorem 14. For every pair , of integers such that 0 < and + 4, there is a connected graph with ( ) = and ( ) = .