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The Connected Edge Monophonic Number of a Graph

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(1)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).

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

(3)  ,

(4)  ,

(5)  ,

(6)   and 

(7)  ,

(8)  ,

(9)  ,

(10)   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

(11) in a connected graph  is said to be a semi-simplicial vertex of  if ∆ 

(12)   |

(13) |  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

(14) be a semisimplicial vertex of . Suppose that

(15)  . Let  be a vertex of  

(16)   such that deg

(17)   |

(18) |  1. Let  ,  , … ,    2 be the neighbors of  in  

(19)  . Since  is a connected edge monophonic set of , the edge 

(20) lies on the monophonic path !: #, # , …  , ,

(21) ,  , … , $, where #, $ % . Since

(22) 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).

(23) 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,

(24) be a cut vertex of  and let  be a connected edge monophonic set of . Then every component of  

(25) contains an element of . Proof. Let

(26) be a cut vertex of  and  be a connected edge monophonic set of . Suppose there exists a component, say  of  

(27) 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 !: ,  ,  , … ,

(28) , … , #, $, …,

(29)  , … ,

(30) , … , '. Since

(31) is a cut vertex of , the   # and $  ' sub paths of P both contain

(32) 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

(33) be any cut vertex of  and let  ,  , … ,  (  2 be the components of  

(34) . 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

(35) % . ∎ Corollary 2.8. For a connected graph  with  semi-simplicial vertices and + cut vertices,    max 2,  / +. 133. 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

(36)  ,

(37)  , … ,

(38)  ,

(39)  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). ≠.

(40) 134. 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  ,

(41)  be a minimum connected edge monophonic set of . Then 

(42) is an edge. If  = 2 , then there exists a vertex w different from  and

(43) . Since 

(44) is a chord,  

(45) 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).

(46) 135. 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

(47) in # such that 7,

(48)  2. Clearly # lies on a  

(49) 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 :

(50)  ,

(51)  , … ,

(52)   be a minimum independent cut set of vertices of . Let  ,  , … ,    2 be the components of   : and let .   :. Then every vertex

(53)  1 ) < ) *  1 is adjacent to at least one vertex of  for any D1 ) D ) . Let 

(54) be an edge of . If uv lies in one of  for any D1 ) D ) , then clearly 

(55) lies on the monophonic path 

(56) itself) joining two vertices  and

(57) of . Otherwise 

(58) is of the form

(59)   1 ) < ) *, where  %  for some D such that 1 ) D ) . As  2,

(60)  is adjacent to some ' in  for some E = D such that. 1 ) E ) . Thus

(61)   lies on the monophonic path ,

(62)  , '. Thus  is an edge monophonic set of , such that    is not connected. However 8 #, #  : is a connected edge monophonic set of  so that ଵ௖   |  

(63) |      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

(64)  ,

(65)  , … ,

(66)  and join each

(67)  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

(68)  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.

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

(70)   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

(71)  ,

(72)  , … ,

(73)  }. 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

(74)  ,

(75)  … ,

(76)  , ' , ' , … , ' to ! and joining each

(77)  1 ) * ) G  F with  ,  ,  and joining each ' 1 ) * ) F  2 with

(78)  . 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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