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29URAL MATHEMATICAL JOURNAL, Vol. 5, No. 2, 2019, pp. 55-63
DOI: 10.15826/umj.2019.2.005
RESTRAINED DOUBLE MONOPHONIC NUMBER OF A GRAPH1
A.P. Santhakumaran2
Department of Mathematics, Hindustan Institute of Technology and Science,
Chennai - 603 103, India [email protected]
K. Ganesamoorthy
Department of Mathematics, Coimbatore Institute of Technology, Coimbatore - 641 014, India [email protected]
Abstract: For a connected graph G of order at least two, a double monophonic set S of a graph G is a restrained double monophonic set if either S = V or the subgraph induced by V — S has no isolated vertices. The minimum cardinality of a restrained double monophonic set of G is the restrained double monophonic number of G and is denoted by dmr (G). The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers a, b, c with 3 < a < b < c, there is a connected graph G with m(G) = a, mr(G) = b and dmr(G) = c, where m(G) is the monophonic number and mr(G) is the restrained monophonic number of a graph G.
Keywords: Monophonic set, Restrained monophonic set, Restrained monophonic number, Restrained double monophonic set, Restrained double monophonic number.
AMS Subject Classification: 05C12.
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 [5]. 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 [2]. The neighborhood of a vertex v is the set N(v) consisting of all vertices u which are adjacent with v. 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 C V, I[S] = U I[x,y]. A set S of vertices of G is a geodetic set if I[S] = V, and the minimum
x,y£S
cardinality of a geodetic set is the geodetic number g(G). The geodetic number of a graph was introduced in [2, 6] and further studied in [3, 4]. It was shown in [6] that determining the geodetic number of a graph is an NP-hard problem. A geodetic set S of a graph G is a restrained geodetic set if the subgraph G[V — S] induced by V — S has no isolated vertex. The minimum cardinality of a restrained geodetic set of G is the restrained geodetic number of G. The restrained geodetic
*The second author research work was supported by National Board for Higher Mathematics, INDIA (Project No. NBHM/R.P.29/2015/Fresh/157).
2Former Professor
number of a graph was introduced and studied in [1]. Let 2V denote the set of all subsets of V. The mapping I: V x V ^ 2V defined by I[u, v] = {w € V : w lies on a u — v geodesic in G} is the interval function of G. One of the basic properties of I is that u,v € I[u, v] for any pair u,v € V. Hence the interval function captures every pair of vertices and so the problem of double geodetic sets is trivially well-defined while it is clear that this fails in many graphs already for triplets (for example, complete graphs). This motivated us to introduce and study double geodetic sets in [7] and further double monophonic sets in [9]. Also, double monophonic parameters like the upper double monophonic number of a graph and the connected double monophonic number of a graph were studied in [10, 11]. This is the basis behind the introduction and study of the restrained double monophonic number of a graph. A set S of vertices is called a double geodetic set of G if for each pair of vertices x, y in G there exist vertices u,v € S such that x,y € I[u, v]. The double geodetic number dg(G) is the minimum cardinality of a double geodetic set. The double geodetic number of a graph was introduced and studied in [7].
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, y € 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 was studied and discussed in [8]. A set S of vertices of G is called a double monophonic set of G if for each pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u — v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set of G. The concept of double monophonic number of a graph was introduced and studied in [9].
The concept of distance in graphs is a major component in graph theory with its centrality and convexity concepts having numerous applications to real life problems. There are several interesting applications of these concepts to facility location in real life situations, routing of transport problems and communication network designs. As the paths involved in the discussion of this paper are monophonic, no intervention by hackers or enemies is possible to the respective facilities provided. Further, as monophonic paths are secured and longer than geodesic paths, it is advantageous to more customers in getting the service with protection.
The following theorems will be used in the sequel.
Theorem 1 [8]. Each extreme vertex of a connected graph G belongs to every monophonic set of G.
Theorem 2 [8]. For the complete graph Kp(p > 2), m(Kp) = p.
Theorem 3 [7]. Each extreme vertex of a connected graph G belongs to every double geodetic set of G.
Throughout this paper G denotes a connected graph with at least two vertices.
1. Restrained double monophonic number
To study the main concepts of this paper, we introduce first the restrained monophonic number of a graph and the restrained double geodetic number of a graph, and then prove some basic results and proceed.
Definition 1. 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 mr (G).
Example 1. For the cycle C5 : u,v,w,x,y,u of order 5, it is easily verified that S = {u,w} is a minimum monophonic set of C5 and so m(C5) = 2. Since the subgraph induced by V — S has an isolated vertex v, S is not a restrained monophonic set of C5. It is clear that, S U {v} is a minimum restrained monophonic set of C5 so that mr(C5) = 3. Thus the monophonic number and the restrained monophonic number of a graph are different.
It is clear that every restrained monophonic set of G is a monophonic set of G and so Theorem 1 gives the next result.
Theorem 4. Each extreme vertex of a connected graph G belongs to every restrained monophonic set of G.
Corollary 1. For the complete graph Kp(p > 2), mr(Kp) = p.
Definition 2. A double geodetic set S of a graph G is a restrained double geodetic set if
either S = V or the subgraph induced by V — S has no isolated vertices. The minimum cardinality of a restrained double geodetic set of G is the restrained double geodetic number of G and is denoted by dgr (G).
Example 2. For the cycle C4 of order 4, it is clear that any set S of two non-adjacent vertices of C4 is a minimum double geodetic set of C4 and so dg(C4) = 2. Since the subgrph induced by V — S has an isolated vertices, S is not a restrained double geodetic set of C4. Also, no 3-element subset of V(C4) is a restrained double geodetic set of C4. Thus V(C4) is the unique minimum restrained double geodetic set of C4 and so dgr (C4) = 4. Hence the double geodetic number and the restrained double geodetic number of a graph are different.
It is clear that every restrained double geodetic set of G is a double geodetic set of G and so Theorem 3 gives the next result.
Theorem 5. Each extreme vertex of a connected graph G belongs to every restrained double geodetic set of G.
Now, we introduce the restrained double monophonic number of a graph and investigate.
Definition 3. A double monophonic set S of a graph G is a restrained double monophonic
set if either S = V or the subgraph induced by V — S has no isolated vertices. The minimum cardinality of a restrained double monophonic set of G is the restrained double monophonic number of G and is denoted by dmr (G).
Example 3. For the graph G given in Fig. 1, it is clear that no 2-element subset of vertices of G is a monophonic set of G. Since S = {v, x, z} is a monophonic set of G, m(G) = 3. It is the only minimum monophonic set of G. Since the subgraph induced by V — S has the isolated vertex v4, it is not a restrained monophonic set of G. Now, the set S1 = {v,x,z,v4} is a restrained monophonic set of G so that mr(G) = 4.
It is easily seen that no 4-element subset of vertices of G containing the vertex v is a double monophonic set of G. Also, it is clear that the set S2 = {v,v1,v2,v3,v4} is the unique minimum double monophonic set of G. Since the subgraph induced by V — S2 has no isolated vertices, S2 is a minimum restrained double monophonic set of G so that dmr (G) = 5. Thus the monophonic number, the restrained monophonic number and the restrained double monophonic number of a graph are different.
v
Theorem 6. Every extreme vertex of a connected graph G belongs to every restrained double monophonic set of G. In particular, if the set of all extreme vertices of G is a restrained double monophonic set, then it is the unique minimum restrained double monophonic set of G.
Proof. Since every restrained double monophonic set is a monophonic set, the result follows from Theorem 1. □
The following results are easy consequences of Theorem 6.
Result 1. For the complete graph Kp(p > 2), dmr(G) = p.
Result 2. For a graph G of order p with k extreme vertices, max{2,k} < dmr (G) < p.
Result 3. If T is a tree of order p with k end-vertices and p — k > 2, then dmr(T) = k.
Theorem 7. For any graph G of order p,
2 < m(G) < mr(G) < dmr(G) < p, mr(G) = p — 1 = dmr(G).
Proof. Any monophonic set needs at least two vertices and hence m(G) > 2. Since every restrained monophonic set is also a monophonic set of G, it follows that m(G) < mr(G). It is clear that every restrained double monophonic set of G is also a restrained monophonic set and so mr(G) < dmr(G). Since the set of all vertices of G is a restrained double monophonic set of G, dmr (G) < p. From the definitions of restrained monophonic number and the restrained double monophonic number, it is clear that mr (G) = p — 1 = dmr (G). □
Remark 1. The bounds in Theorem 7 are sharp. The two end-vertices of a nontrivial path Pn on n vertices is its unique minimum monophonic set so that m(Pn) = 2 and for the complete graph Kp(p > 2), we have dmr(Kp) = p. Also, all the inequalities in Theorem 7 can be strict, for the graph G of order 8 given in Fig. 1, m(G) = 3, mr(G) = 4 and dmr(G) = 5. Thus we have 2 < m(G) < mr(G) < dmr(G) < p.
Theorem 8. For any graph G of order p, 2 < dmr (G) < dgr (G) < p, dmr(G) = p — 1 = dgr (G).
Proof. Any restrained double monophonic set needs at least two vertices and so dmr (G) > 2. It is clear that every restrained double geodetic set of G is also a restrained double monophonic set and so dmr (G) < dgr (G). Since the set of all vertices of G is a restrained double geodetic set of G, dgr (G) < p. From the definitions of restrained double monophonic number and the restrained
double geodetic number, it is clear that dmr(G) = p — 1 = dgr (G). □
Remark 2. The bounds in Theorem 8 are sharp. For the path Pn(n > 4), dmr(G) = dgr (G) = 2 and for the complete graph Kp(p > 3), dmr(Kp) = dgr(Kp) = p. All the inequalities in Theorem 8 can be strict. For the graph G of order 7 given in Fig. 2, no 2-element subset of V(G) forms a minimum restrained double monophonic set of G. The minimum restrained double monophonic sets of G are Si = {v^v2,v5} and S2 = {v^v2,v6} so that dmr(G) = 3. Also, there is no 3-element or 4-element subset of V(G) forms a minimum restrained double geodetic set of G. It is easy to verify that S3 = {v1,v2,v4,v5,v6} is a minimum restrained double geodetic set of G and so dgr(G) = 5. Thus we have 2 < dmr (G) < dgr (G) < p.
Vi V2
Figure 2. Graph G.
The following results are easy to prove.
Result 4. For any cycle Cp, dmr(Cp) =
p, if p = 3, 4, 3, if p = 5, 2, if p > 6.
4,
if p = 4, if p > 5.
Result 5. For any wheel Wp = K1 + Cp-1, (p > 4), dmr (Wp) =
Result 6. For the complete bipartite graph G = Km,n(2 < m < n),
dmr(G) = (n + 2 if 2 = m < n, I 4, if 3 < m < n.
In view of Theorem 7, we have the following realization theorem.
Theorem 9. For any three integers a, b, c with 3 < a < b < c, there is a connected graph G with m(G) = a, mr(G) = b and dmr(G) = c.
Proof. This theorem is proved by considering four cases. Case 1. a = b = c. Then, for the complete graph G = Ka, by Theorem 2, Corollary 1 and Result 1, m(G) = mr(G) = dmr(G) = a.
Case 2. a = b < c. Let G be the graph in Fig. 3 is got by adding a — 1 new vertices w1 ,w2, ■ ■ ■ , wa-2,x to the complete bipartite graph K2,c-a+1 with the partite sets U = {u1,u2} and W = {v1 ,v2, ■ ■ ■ , vc-a+1}, joining each vertex w^(1 < i < a — 2) to the vertex u1 and joining the vertex x to the vertex v1. By Theorems 1, 4 and 6, every monophonic set, every restrained monophonic
set and every restrained double monophonic set of G contain the set S = {w1,w2, ■■■ ,wa-2,x} of all extreme vertices of G. Clearly, S is not a monophonic set of G. It is easy to verify that S1 = S U {u2} is the unique minimum monophonic set of G and so m(G) = a. Since the subgraph induced by V — S1 has no isolated vertices, S1 is the unique minimum restrained monophonic set of G so that mr (G) = a = m(G). It is clear that the pair of vertices x, v-i (i = 2,3, ■ ■ ■ ,c — a + 1) do not lie on any u — v monophonic path, for any u,v € S1 and so S1 is not a restrained double monophonic set of G. It is easy to verify that S2 = S U {u2,v2, ■ ■ ■ ,vc-a+1} is a minimum double monophonic set of G and the subgraph induced by V — S2 has no isolated vertices so that dmr (G) = c.
Wi W2 Wa-2
Figure 3. Graph G.
Case 3. a < b = c. Let G be the graph in Fig. 4 formed from the path P3 : v1,v2,v3 of order 3, by adding b new vertices u1,u2, ■ ■ ■ , ua-2,w1,w2,..., wb-a+2 to P3 and joining each vertex ui(1 < i < a — 2) to v2; and joining each vertex wj(1 < j < b — a + 2) to v1 and v3. By Theorems 1, 4 and 6, every monophonic set, every restrained monophonic set and every restrained double monophonic set of G contain the set S = {u1 ,u2, ■ ■ ■ ,ua-2} of all extreme vertices of G. Clearly, S is not a monophonic set of G. Also, for any x € V — S, S U {x} is not a monophonic set of G. It is easy to verify that S1 = S U {v1,v3} is a minimum monophonic set of G and so m(G) = a. Since the subgraph induced by V — S1 has the isolated vertices w1,w2, ■ ■ ■ ,wb-a+2,v2,
51 is not a restrained monophonic set of G. It is clear that, every restrained monophonic set and every restrained double monophonic set of G contains {w1,w2, ■■■ ,wb-a+2} and it follows that
52 = S U{w1,w2, ■ ■ ■ , wb-a+2} is a minimum restrained monophonic set and a minimum restrained double monophonic set of G so that mr (G) = dmr (G) = b.
Ui U2 Ua-2
Case 4. a < b < c. Let G be the graph in Fig. 5 formed from the path P3 : x,y,z of order 3 by adding c new vertices u1,u2,..., ua-2,w1 ,w2,..., wb-a, v1,v2,..., vc-b+2 to P3 and joining each vertex vi(1 < i < c — b + 2) to the vertices x, y and z; joining each vertex wj(1 < j < b — a) to the
vertices x and z; joining each vertex u^(1 < k < a—2) to the vertex y. By Theorems 1, 4 and 6, every monophonic set, every restrained monophonic set and every restrained double monophonic set of G contain the set S = {u1,u2, ■ ■ ■ ,ua-2} of all extreme vertices of G. Clearly, S is not a monophonic set of G and also for any u € V(G) — S, S U {u} is not a monophonic set of G. It is easily verified that S1 = S U {x, z} is a minimum monophonic set of G and so m(G) = a. Since the subgraph induced by V — S1 has the isolated vertices w2,w3, ■ ■ ■ ,wb-a, S1 is not a restrained monophonic set of G. It is clear that every restrained monophonic set of G contains {w2,w3, ■ ■ ■ ,wb-a}. Then S2 = S1 U {w1, w2,..., wb-a} is a minimum restrained monophonic set of G and so mr(G) = b.
Now, any double monophonic set of G should contain the set S. It is easily verified that the set S' = {u1, u2, ■ ■ ■ , ua-2,w1,w2,..., wb-a, v1,v2,..., vc-b+2} is the unique minimum double monophonic set of G. Since the subgraph induced by V — S' has no isolated vertices, it follows that S' is a minimum restrained double monophonic set of G so that dmr (G) = c. □
Figure 5. Graph G.
Theorem 10. For any integer p > 4 with 2 < k < p, k = p — 1, there is a connected graph G of order p such that dmr (G) = k.
Proof. Let G = K1p-1 be any tree of order p with k end-vertices. Then clearly, dmr (G) = k. □
In view of Theorem 8, we have the following realization theorem.
Theorem 11. For every pair a, b of integers with 3 < a < b, there is a connected graph G with dmr (G) = a and dgr (G) = b.
Proof. For 3 < a = b, by Theorem 5 and Result 1, the complete graph Ka of order a has the desired properties. So, assume that 3 < a < b. Let H be the graph obtained from the complete graph Ka-1 and the path P4 : u,v,w,x of order 4 by joining all the vertices of Ka-1 to the vertex u of P4. Let G be the graph in Fig. 6 obtained from H by taking 'b — a — 1' copies of the path Pi : xi, yi, zi(1 < i < b — a — 1) of order 3 and joining each vertex x^(1 < i < b — a — 1) in Pi and u in H; and also joining each vertex zi(1 < i < b — a — 1) in Pi and w in H. Let S = V(Ka-1) U {x} be the set of all extreme vertices of G. By Theorems 5 and 6, every restrained double geodetic set and every restrained double monophonic set of G contain S. Clearly, S is the unique minimum restrained double monophonic set of G and so dmr (G) = a. Since the pair of vertices v,yi (i = 1,2,3, ■ ■ ■ ,b — a — 1), do not lie on any geodesic joining a pair of vertices from S, S is not a restrained double geodetic set of G. Let S' = S U {v,y1,y2, ■ ■ ■ ,yb-a-1}. It is easy to verify that S' is a minimum double geodetic set of G and the subgraph induced by V — S' has no isolated vertices. Thus S' is a minimum restrained double geodetic set of G and so dgr(G)=b. □
Vb-a-1
Figure 6. Graph G.
Theorem 12. If G' is a graph obtained by adding k pendant vertices to a connected graph G, then dmr(G) < dmr(G ) < dmr(G) + k.
Proof. Let G' be the connected graph obtained from G by adding k pendant vertices vi(1 < i < k) to the vertices ui(1 < l < k) where each ui is a vertex of G and each vi(1 < i < k) is not a vertex of G. Note that u1,u2, ■ ■ ■ ,uk need not be distinct. Let S be a minimum restrained double monophonic set of G. Then S U {v1,v2, ...,vk} is a restrained double monophonic set of G and so dmr(G ) < dmr(G) + k.
Now, we claim that dmr (G) < dmr (G ). Suppose that dmr (G) > dmr (G ). Let S be a restrained double monophonic set of G with IS | < dmr(G). Since each v,i (1 < i < k) is an extreme vertex of G , it follows from Theorem 6 that {v1,v2, ...,vk} C S . Let S = (S — {v1,v2,...,vk}) U{u1,u2,...,ui} (1 < l < k). Then S is a subset of V(G) and IS I < |S'| — k +1 = IS'I — (k — l) < |S | < dmr (G). Now, we show that S is a restrained double monophonic set of G. Let u,v € V(G) — S. Then u,v € V(G') also. Since S is a restrained double monophonic set of G , u and v lie on a x — y monophonic path P in G' for some vertices x,y € S . If neither x nor y is vi(1 < i < k), then x,y € S. If exactly one of x, y is vi(1 < i < k), say x = vi, then u and v lie on a x — y monophonic path in G, where y = uj and uj is adjacent to vj in G' where i = j. If both x,y € {v1,v2, ...,vk}, then let x = vi and y = vj where i = j. Hence u and v lie on the us — ut monophonic path in G, where us is adjacent to v,i and ut is adjacent to vj in G'. Thus S is a restrained double monophonic set of G. Hence dmr(G)<|SI<dmr(G), which is a contradiction. □
Remark 3. The bounds for dmr (G') in Theorem 12 are sharp. Consider a tree T with number of end-vertices l > 3 and at least two internal vertices. Let S = {v1,v2, ...,vl} be the set of all end-vertices of T. Then by Result 3, dmr (T) = l. If we add a pendant vertex to an end-vertex of T, then we obtain another tree T with l end-vertices. Hence dmr(T) = dmr(T ). On the other hand, if we add k pendant vertices to a cut-vertex of T, then we obtain a tree T with k +1 end-vertices. Then by Result 3, dmr (T') = dmr (T) + k.
2. Conclusions
In this paper, the concept of restrained double monophonic number of a graph is introduced and certain general properties satisfied by this parameter are studied. This parameter is determined for several standard graphs. Also, certain realisation results of this parameter are proved with regard to certain other parameters like monophonic number, restrained monophonic number and restrained double geodetic number of a graph. As a future work of this paper, new parameters like
upper restrained double monophonic number of a graph, forcing restrained double monophonic number of a graph can be developed and investigated.
Acknowledgement
The authors are thankful to the reviewers for their useful comments for the improvement of this paper.
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