Cmjv03i02p0131

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J. Comp. & Math. Sci. Vol.3 (2), 131-136 (2012)

The Connected Edge Monophonic Number of a Graph J. JOHN1, P. ARUL PAUL SUDHAHAR2 and A. VIJAYAN3 1

Department of Mathematics Government College of Engineering, Tirunelveli - 627007, India. 2 Department of Mathematics Alagappa Government Arts College, Karaikudi – 630 003, India. 3 Department of Mathematics, N.M. Christian College, Marthandam- 629001, India. (Received on: February 25, 2012) ABSTRACT For a connected graph G = (V, E), a set ‫ܸ ك ܯ‬ሺ‫ܩ‬ሻ is called a connected edge monophonic set if the subgraph ‫ܩ‬ሾ‫ܯ‬ሿ induced by ‫ ܯ‬is connected. The minimum cardinality of a connected edge monophonic set of ‫ ܩ‬is the connected edge monophonic number of ‫ ܩ‬and is denoted by ݉ଵ௖ ሺ‫ܩ‬ሻ. Connected graphs of order p with connected edge monophonic number 2 and p are characterized. It is shown that for every two integers a, b and c such that 2 ≤ a < b < c, there exists a connected graph G with m(G)=a, m1(G)=b and m1c(G) where m(G) is the monophonic number and m1(G) is the edge monophonic number of G. Keywords: monophonic path, monophonic number, monophonic number, connected edge monophonic number.

edge

AMS Subject Classification : 05C05.

1. 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 Harary1. A chord of a path u0, u1, u2, …, uh is an edge uiuj, with j ≥ i + 2. An u-v path is called a monophonic path if it is a

chordless path. A monophonic set of G is a set M ⊆ V(G) 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 of G. The monophonic number of a graph G is studied in2,3,4. The maximum degree of G, denoted by ∆(G), is given by ∆(G) = max{degG(v):v ∈ V(G)}. N(v) = {u ∈ V(G):

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

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uv ∈ E(G)} is called the neighborhood of the vertex v in G. For any set S of vertices of G, the induced subgraph <S> is the maximal subgraph of G with vertex set S. A vertex v is an extreme vertex of a graph G if <N(v) > is complete. A connected monophonic set of a graph G is a monophonic set M such that the subgraph <M> induced by M is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by mc(G). A connected monophonic set of cardinality mc(G) is called a mc-set of G or a minimum connected monophonic set of G. The connected monophonic number of a graph is studied in5. An edge monophonic set of G is a set M ⊆ V(G) such that every edge of G is contained in a monophonic path joining some pair of vertices in M. The edge monophonic number m1(G) of G is the minimum order of its edge monophonic sets and any edge monophonic set of order m1(G) is a minimum edge monophonic set of G. The edge monophonic number of a graph is studied in6. 2. THE CONNECTED EDGE MONOPHONIC NUMBER OF A GRAPH Definition 2.1. A set is called a connected edge monophonic set if the subgraph induced by is connected. The minimum cardinality of a connected edge monophonic set of is the connected edge monophonic number of and is denoted by . A connected edge monophonic set of size is said to be a -set. Example 2.2. For the graph given in Figure 2.1, , , , and , , , are the only two

connected edge monophonic sets of so that 4.

v1 v7

v2

v6

v3

v5

v4 G Figure 2.1

Definition 2.3. A vertex in a connected graph is said to be a semi-simplicial vertex of if ∆ | | 1. Theorem 2.4. Each semi-simplicial vertex of a graph belongs to every connected edge monophonic set of . Proof. Let be a connected edge monophonic set of . Let be a semisimplicial vertex of . Suppose that . Let be a vertex of such that deg | | 1. Let , , ‌ , 2 be the neighbors of in . Since is a connected edge monophonic set of , the edge lies on the monophonic path !: #, # , ‌ , , , , ‌ , $, where #, $ % . Since is a semi – simplicial vertex of , and are adjacent in and so P is not a monophonic path of , which is a contradiction.∎ Corollary 2.5. Each simplicial vertex of a graph belongs to every connected edge monophonic set of .

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

J. John, et al., J. Comp. & Math. Sci. Vol.3 (2), 131-136 (2012)

Poof. Since every simplicial vertex of is a semi-simplicial vertex of , the result follows from Theorem 2.4.∎ Theorem 2.6. Let be a connected graph,

be a cut vertex of and let be a connected edge monophonic set of . Then every component of contains an element of . Proof. Let be a cut vertex of and be a connected edge monophonic set of . Suppose there exists a component, say of such that contains no vertex of . By Corollory 2.5, contains all the simplicial vertices of and hence it follows that does not contains any simplicial vertex of . Thus contains at least one edge, say #$. Since is a connected edge monophonic set, #$ lies on the ' monophonic path !: , , , ‌ , , ‌ , #, $, ‌,

, ‌ , , ‌ , '. Since is a cut vertex of , the # and $ ' sub paths of P both contain and so ! is not a monophonic path, which is a contradiction.∎ Theorem 2.7.Each cut vertex of a connected graph belongs to every minimum connected edge monophonic set of . Proof. Let be any cut vertex of and let , , ‌ , ( 2 be the components of . Let be any connected edge monophonic set of . Then by Theorem 2.6, contains at least one element from each : 1 ) * ) ( . Since is connected, it follows that % . ∎ Corollary 2.8. For a connected graph with semi-simplicial vertices and + cut vertices, max 2, / +

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Proof. This follows from Theorems 2.5 and 2.7. ∎ In the following we determine the connected edge monophonic member of some standard graphs. Corollary 2.9. i) For any non-trivial tree 0 of order 1, 0 1 / 1. ii) For the complete graph 2 1 2 , 32 4 1 Theorem 2.10. 3 , 32 4 3

For the cycle 5 1

Proof. Let , , ‌ , , be a cycle of length 1. Let #, $ % 5 such that 7 #, $ 2. Then #, $ is an edge monophonic set of 5 . But is not connected. Let be a vertex of 5 which is adjacent to both # and $. Then 8 is a connected edge monophonic set of so that 35 4 3.∎ Theorem 2.11. For the complete bipartite graph 2 , , (i) (G) = 2 if m = n = 1. (ii) (G) = n+1 if m = 1, n ≼ 2. (iii) (G) = min{m,n}+1, if m, n ≼ 2. Proof. i) This follows from Corollary 2.9 (ii) ii) This follows from Corollary 2.9 (i) iii) Let , 9 2. First assume that 9. Let : , , ‌ , and ;

' , ' , ‌ , ' be a partition of . Let M=U ⋃ {w1}. We prove that is a minimum connected edge monophonic set of . Any

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

â‰

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J. John, et al., J. Comp. & Math. Sci. Vol.3 (2), 131-136 (2012)

edge ' 1 ) * ) , 1 ) < ) 9 lies on the monophonic path ' for any = * so that is an edge monophonic set of . Since is connected, is a connected edge monophonic set of . Let 0 be any set of vertices such that |0| | |. If 0 > :, 0 is not connected and so 0 is not a connected edge monophonic set of . If 0 > ', again 0 is not a connected edge monophonic set of by a similar argument. If 0 ? :, then since |0| |@|, we have 0 :, which is not a connected edge monophonic set of . Similarly, since |0| |@|, 0 cannot contain ;. For if T ⊇ W, then |0| 9 / 1 | |,which is a contradiction. Thus 0 A : 8 ; such that 0 contains at least one vertex from each of : and ;. Then since |0| | |, there exists vertices % : and ' % ; such that 0 and ' ∉ T. Then clearly ' does not lie on a monophonic path connecting two vertices of 0 so that 0 is not a connected edge monophonic set of . Thus in any cases T is not a connected edge monophonic set of . Hence is a minimum connected edge monophonic set of so that 2 , / 1 . Now, if 9, we can prove similarly that : 8 $ , where $ % ; is a minimum connected edge monophonic set of . Hence the theorem follows. ∎ Theorem 2.12. For any connected graph of order , 2 ) ) ) 1 . Proof. Any edge monophonic set needs at least two vertices and so 2. Since every connected edge monophonic set is also an edge monophonic set, it follows that ) . Also, since

induces a connected edge monophonic set of , it is clear that ) 1. ∎ Remark 2.13. The bounds in theorem 2.12 are sharp. For any non-trivial path !, ! 2. For the complete graph 2 , 32 4 32 4 1. Also, all the in equalities in the theorem are strict. For the graph given in Figure 2.2,

3, 5 and 1 7 so that 2 1.

v4

v5 v1

v6 v7

v3

v2 G Figure 2.2

The following Theorems 2.14 and 2.15 characterize graphs for which 2 and 1, respectively. Theorem 2.14. Let be a connected graph of order 1 2 . Then 2 if and only if 2. Proof. Let 2 , then 2. Conversely, let 2. Let , be a minimum connected edge monophonic set of . Then is an edge. If = 2 , then there exists a vertex w different from and

. Since is a chord, is not a monophonic path so that is not a set, which is a contradiction. Thus 2 .∎

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

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J. John, et al., J. Comp. & Math. Sci. Vol.3 (2), 131-136 (2012)

Theorem 2.15. Let be a connected graph. Then every vertex of is either a cut vertex or a semi-simplicial vertex if and only if 1. Proof. Let be a connected graph with every vertex of is either a cut vertex or a semi-simplicial vertex. Then the result follows from Theorems 2.4 and 2.6. Conversely, let 1. Suppose that there is a vertex # in which is neither a cut vertex nor a semi-simplicial vertex. Since # is not a semi-simplicial vertex, # does not induce a complete sub graph and hence there exists and in # such that 7 , 2. Clearly # lies on a

monophonic path in . Also, since # is not a cut vertex of G, # is connected. Thus # is a connected edge monophonic set of and so ) | # | 1 1, which is a contradiction. ∎ Theorem 2.16. If is a non-complete connected graph such that it has a minimum cut set of consisting of * independent vertices, then ) 1 * / 1. Proof. Since is non-complete, it is clear that 1 ) * ) 1 2. Let : , , ‌ , be a minimum independent cut set of vertices of . Let , , ‌ , 2 be the components of : and let

:. Then every vertex 1 ) < ) * 1 is adjacent to at least one vertex of for any D 1 ) D ) . Let be an edge of . If uv lies in one of for any D 1 ) D ) , then clearly lies on the monophonic path itself) joining two vertices and

of . Otherwise is of the form 1 ) < ) * , where % for some D such that 1 ) D ) . As 2, is adjacent to some ' in for some E = D such that

1 ) E ) . Thus lies on the monophonic path , , '. Thus is an edge monophonic set of , such that is not connected. However 8 # , # : is a connected edge monophonic set of so that ଵ௖ | | 1. ∎ Theorem 2.17. For every pair , 1 of integers with 3 ) ) 1, there exists a connected graph of order 1 such that . Proof. Let ! : , , ‌ , be a path on k vertices. Add new vertices , , ‌ , and join each 1 ) * ) 1 with and , there by obtaining the graph in Figure 2.3. Then has order 1 and

, ‌ , is the set of all cut vertices and simplicial vertices of . By Corollary 2.5 and Theorem 2.7, 2 . clearly is not a connected edge monophonic set of and so 2. Now, either 8 1 ) * ) 1 2 nor 8 is an edge monophonic set of . But 0 8 is an edge monophonic set of such that 0 is disconnected. It is clear that 08 is a connected edge monophonic set of and hence it follows that . u1

u2

u3

u4

v1 v2

vp-k G Figure 2.3

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

uk

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J. John, et al., J. Comp. & Math. Sci. Vol.3 (2), 131-136 (2012)

, , ‌ , , be the set of all simplicial vertices of . It is clear that is not a monophonic set of . However 8 is a monophonic set of so that F. Let 8 , , ‌ , }. It is clear that is an edge monophonic set of so that F / G F G. It can be easily varied that is not a connected edge monophonic set of . However

: , , ‌ , is a connected edge monophonic set of so that G / H G H. ∎

Theorem 2.18. For any positive integers 2 ) F G H, there exists a connected graph such that F,

G and H. Proof. Let be the graph given in Figure 2.4 obtained from the path on H G / 2 vertices ! : , , ‌ , by adding G 2 new vertices

, ‌ , , ' , ' , ‌ , ' to ! and joining each 1 ) * ) G F with , , and joining each ' 1 ) * ) F 2 with . Let

w

w

wa-2 u1

u3

u2

u4

uc-

v1 v2

vb-a

G Figure 2.4

REFERENCES 1. F. Buckley and F. Harary, Distance in Graphs, Addition- Wesley, Redwood City, CA, (1990). 2. Esamel M. Paluga, Sergio R. Canoy, Jr., Monophonic numbers of the join and Composition of connected graphs, Discrete Mathematics 307, 1146 – 1154 (2007). 3. Mitre C. Dourado, Fabio protti and Jayme L. Szwarcfiter, Algorithmic Aspects of Monophonic Convexity,

Electronic Notes in Discrete Mathematics 30, 177-182 (2008). 4. J. John, S. Panchali, The upper monophonic number of a graph, International. Math. Combin,4, 46-52 (2010). 5. J. John, P. Arul Paul Sudhahar, The connected monophonic number of a graph, International Journal of Combinatorial Graph Theory and Application (in press). 6. J. John, P. Arul Paul Sudhahar, On the edge monophonic number of a graph (Submitted).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

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