Arithmetic ODD Decomposition of Extented Lobster

Transcription

1 Global Journal of Mathematical Sciences: Theory and Practical. ISSN Volume 4, Number 1 (01), pp International Research Publication House Arithmetic ODD Decomposition of Extented Lobster 1 E. Ebin Raja Merly and N. Gnanadhas 1 Department of Mathematics, Nesamony Memorial Christian College, Marthandam Kanyakumari District, Tamil Nadu, India ebinmerly@gmail.com Former Principal, Nesamony Memorial Christian College, Marthandam Kanyakumari District, Tamil Nadu, India n.gnanadhas@gmail.com Abstract Let G = (V, E) be a simple connected graph with p vertices and q edges. If G 1,G,,G n are connected edge disjoint subgraphs of G with E(G)=E(G 1 ) E(G ) E(G n ), then (G 1, G,, G n ) is said to be a decomposition of G. A decomposition (G 1, G,, G n ) of G is said to be continuous monotonic decomposition(cmd) if each G i is connected and E(G i ) =i, for every i = 1,, 3,,n. In this paper, we introduced the concept arithmetic odd Decomposition. A decomposition (G 1, G,, G n ) of G is said to be a Arithmetic Decomposition or Linear decomposition if E(G i ) = a+(i- 1)d, for every i=1,, 3,, n and a,d Z. Clearly n q = [ a + ( n 1) d ]. If a=1 and d=1, then n ( n + 1) q =.That is, Arithmetic decomposition is a CMD. In this paper, we study the graphs when a=1 and d=. If d=, then q = n. That is, the number of edges of G is a perfect square. Also we obtained the bounds for n and diameter of Extended Lobster L E. Keywords: Decomposition of Graph, Continuous Monotonic Decomposition, Arithmetic Decomposition or Linear Decomposition, Arithmetic Odd Path Decomposition (OPD), Arithmetic Odd Star Decomposition (OSD). AMS Subject Classification: 05C99. Introduction All basic terminologies from Graph Theory are used in this paper in the sense of

2 36 E. Ebin Raja Merly and N. Gnanadhas Harary [3]. By a graph we mean a finite, undirected graph without loops or multiple edges. Definition 1.1: Let G = (V, E) be a simple connected graph with p vertices and q edges. If G 1, G,,G n are connected edge disjoint subgraphs of G with E(G)=E(G 1 ) E (G ) E(G n ), then (G 1, G,,G n ) is said to be a Decomposition of G. Figure (1): Decomposition (G 1, G, G 3 ) of G. N.Gnanadhas and J.Paulraj Joseph discussed on Continuous Monotonic Decomposition (CMD) of graphs [4] and [5]. E.Ebin Raja Merly and N.Gnanadhas discussed Linear Path Decomposition or arithmetic odd path decomposition of Lobster [1] and Linear star decomposition or arithmetic odd star decomposition of Lobster []. This paper deals with Arithmetic odd Decomposition for a very particular class of unicyclic graph namely Extended Lobster denoted by L E. Definition 1.: A Decomposition (G 1, G,, G n ) of G is said to be Continuous Monotonic Decomposition (CMD) if E (G i ) =i, for every i=1,, 3,,n. Clearly n( n + 1) q = Figure (): Continuous Monotonic Decomposition (G 1, G, G 3, G 4 ) of G

3 Arithmetic ODD Decomposition of Extented Lobster 37 Definition 1.3: A decomposition (G 1, G,, G n ) of G is said to be an Arithmetic decomposition or Linear decomposition if E(G i ) = a+ (i-1) d, for every i=1,, 3, n, n and a, d Z. Clearly q = [ a + ( n 1) d ] n( n +1) If a=1 and d=1, then q =. That is, Arithmetic decomposition is a CMD. If a=1 and d = then, q = n. That is, the number of edges of G is a perfect square. Since the number of edges of G is a perfect square, q is the sum first n odd numbers 1, 3, 5,, (n-1). Thus we call the Arithmetic Decomposition with a = 1 and d = as Arithmetic Odd Decomposition (AOD). Since the number of edges of each subgraph of G is odd, we denote the AOD as (G 1, G 3, G 5,, G (n-1) ). Example 1.4: For the graph G in figure (3), (G 1, G 3, G 5, G 7 ) is an AOD. Figure (3) Some definitions will be helpful here. Definition 1.5: Unicyclic graph is a connected graph containing exactly one cycle. Definition 1.6: An Arithmetic odd decomposition (G 1, G 3, G 5,, G n-1 ) in which each G i is a path P i with i edges is said to be an Arithmetic odd Path Decomposition or simply odd path decomposition(opd)

4 38 E. Ebin Raja Merly and N. Gnanadhas Definition 1.7: Caterpillar is a tree in which the removal of pendant vertices results in a path. Definition 1.8: Lobster is a tree in which the removal of pendant vertices results in a caterpillar. Definition 1.9: The underlying path of a Lobster L is a path obtained by the removal of pendant vertices two times successively. ODD Path Decomposition of Extended Lobster Definition.1: Let L be a Lobster with n -1 edges. Then the graph denoted by L E is obtained by adding an edge e to L that forms a unicyclic graph is called an Extended Lobster. Remark.: Extended Lobster is a graph which is not a Lobster. Clearly L E has n edges. Hence L E admits AOD. Let L E be the extended Lobster with q = n. Then L E = L + e where L is the Lobster with underlying path. The unicycle in L E is C k = P k-1 P 1, 3 k n -1. Definition.3: If L E admits decomposition (P 1, P 3, P 5,, P (n-1) ), then the decomposition is called an Arithmetic Odd Path Decomposition (OPD) of L E. Remark.4: For OPD in L E, always we treat P 1 as e. Remark.5: In this paper, we study the Extended Lobster L E with q= and so the term Extended Lobster L E always means L E with q=. Our main theorem can now be stated as follows: Theorem.6: If the extended Lobster L E admits OPD (P 1, P 3, P 5,, P (n-1) ), then 5. Proof: Assume that L E admits OPD. Clearly diam (L E ) 4. Figure (4)

5 Arithmetic ODD Decomposition of Extented Lobster 39 Case (i): P 1 and P 3 can be obtained from L E without taking any edge from. Then, for P 5, we must have only one edge from, for P 7, we must have 3 edges from, for P 9, we must have 5 edges from,, for P (n-1), we must have [( -1)-4] edges from. Hence = [(i-1)-4] + + [( -1)-4] = ( -) = Case (ii) Each path P i-1, i =, 3, 4,, n has edges from. Then, for P 3, we must have one edge from, for P 5, we must have one edge from, for P 7, we must have three edges from, for P 9, we must have five edges from, for P (n-1),we must have [( -1)-4] edges from. Thus = [( -1)-4] = = 1 Case (iii): atleast one edge of each P (i-1), i =, 3, 4,, n must be in. Then = [( -1)-4], which is same as case (ii). Case (iv): Let P (r-1) and P (s-1) be two paths in the decomposition with origin v r and v s respectively. Figure (5) Then we have = (n-1) = - 5 = 5. Hence 5. Remark.7: Let L E be an extended Lobster and be the underlying path obtained from L E e. Let N 1 denotes the set of vertices in L E e which are at a distance one from. Let n 1 = N 1. Let N denotes the set of pendant vertices of L E e which are at a distance two from. Let n = N. Theorem.8: Let L E be the Extended Lobster with underlying path of length and. If L E admits OPD (P 1, P 3, P 5,, P (n-1) ), then n = n-3. Proof: Suppose L admits OPD. Since, no edge of P 1 and P 3 must be in. Thus P 1 contributes 0 for n, P 3 contributes 1 for n, P 5 contributes atmost for n, P 7

6 40 E. Ebin Raja Merly and N. Gnanadhas contributes atmost for n,, P (n-1) contributes atmost for n. Thus n = 1+(n-) = n-3. Example.9: We take L E with q=5. Consider L E e Then n = =9. Figure (6) Here the underlying path of L E e is v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10. Clearly P 1 = e = u 7 w 8.From the Lobster L E e, we can easily construct P 3 as u 4 w 4 v 7 w 8, since no edge of P 3 must be in. Also P 5 is u 1 w 1 v 1 v w u, P 7 is u 3 w 3 v 5 v 4 v 3 v w 6 u 6 and P 9 is u 5 w 5 v 10 v 9 v 8 v 7 v 6 v 5 w 7 u 7. Hence N = {u 1, u, u 3, u 4, u 5, u 6, u 7 } and n =7. Theorem.10: If L E be an extended Lobster with underlying path of length and = 5. Then L E admits OPD (P 1, P 3, P 5,, P (n-1) ) if and only if L E e is a path. Proof: Assume that L E admits OPD. To prove all the internal vertices of vertices of L E e are of degree. Suppose not. Let u be an internal vertex of L E e of degree > as shown in figure (7). Figure (7)

7 Arithmetic ODD Decomposition of Extented Lobster 41 Let e 1 be an edge incident with u, which is not in L E e. Then e 1 is the first or the last edge of some path P (k-1). Thus 4 = (k-3)+(k-1-1)+(k+1)+ +(n-1). 5 = n -1 n = 6 which is a contradiction. Hence L E e is a path. The converse part is obvious, since L E has q=. ODD Star Decomposition of Extended Lobster Definition 3.1: If L E admits decomposition (S 1, S 3, S 5,,S (n-1) ), then the decomposition is known as Arithmetic odd Star Decomposition or simply odd star decomposition(osd). Remark 3.: Let L E be the extended Lobster with q = n. Then L E e is a Lobster with the longest path P. Remark 3.3: For OSD in L E, always we treat S 1 as e. Result 3.4: If L E admits OSD (S 1, S 3, S 5,, S (n-1) ), then diam (L E e) n-. Proof: diam (L E e) diam (S 3 ) + diam (S 5 ) + diam (S 7 ) + + diam (S (n-1) ) = n-. Hence diam (L E e) n-. Now we are ready to prove Theorem 3.5. Theorem 3.5: Let L E be an Extended Lobster with q = n and diam (L E e) = n-. If L E admits OSD (S 1, S 3, S 5,, S (n-1) ) with S 1 = e if and only if L E e is a caterpillar with (n-1) non-adjacent junctions and There is no junction neighbour in L E. Proof: Suppose L E admits OSD. Since diam (L E e) = n-, the centres of S 3, S 5,, S (n-1) lie in P. Thus L E e is a caterpillar. Since S 1 is e and diam (L E e) = n-, there is exactly one non-support in between any two centres. Hence there are (n-1) nonadjacent junctions in L E e. Next to prove there is no junction-neighbor in L E - e. Suppose there is atleast one junction neighbor in L E e. Let the junction neighbor be e i = x i y j. Then there exist junction supports v i and v j such that d (v i, v j ) 3. Therefore < E (L E e) E (S 3 S 5 S n-1 ) > = S 1, which is a contradiction. Hence there is no junction neighbor in L E e. The converse part is obvious. References [1] Ebin Raja Merly, E. and Gnanadhas, N. 011, Linear Path Decomposition of Lobster, International Journal of Mathematics Research. Volume 3, Number 5, pp

8 4 E. Ebin Raja Merly and N. Gnanadhas [] Ebin Raja Merly,E. and Gnanadhas, N. 01, Linear star Decomposition of Lobster, International Journal of Contemporary Mathematical Sciences, Vol.7, no. 6, [3] Frank Harary, 197, Graph theory, Addison Wesley Publishing Company. [4] Gnanadhas N.and Paulraj Joseph J., 003, Continuous Monotonic Decomposition of [5] Cycles, International Journal of Management system, Vol.19, No.1, Jan-April pp [6] Gnanadhas, N. and Paulraj Joseph, J. 000, Continuous Monotonic Decomposition of Graphs, International Journal of Management system. 16(3),