Connected Domination Path Decomposition Polynominal of Path and Cycle
D. Jeya Jothi * and E. Ebin Raja Merly1
*Research Scholar, Department of Mathematics, Nesamony Memorial Christian College, Marthandam – 629165, Tamil Nadu, INDIA.
1Assistant Professor, Department of Mathematics, Nesamony Memorial Christian College, Marthandam – 629165, Tamil Nadu, INDIA.
Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli – 627012, Tamil Nadu, INDIA.
email: [email protected], [email protected]
(Received on: June 7, 2018)
ABSTRACT
Let G = (V,E) be a simple connected graph with p vertices and q edges. A Decomposition (G1, G2 ,...,Gn) of G is said to be connected domination decomposition (CDD) if (i) E(G) = E(G1) ∪ E(G2) ∪ ... ∪ E(Gn) (ii) Each Gi is connected . (iii)
(Gi) = i, 1 ≤ i ≤ n. In this paper, we establish connected domination path decomposition polynomial of a graph G. In particular, we investigate connected domination path decomposition polynomial of path and cycle.
AMS Mathematics Subject Classification (2010): 97k30
Keywords: Connected Domination Decomposition, Connected Domination Path Decomposition, Connected Domination Path Decomposition Polynomial.
1. INTRODUCTION
All basic terminologies from graph theory are used in this paper in the sense of Frank Harary3. By a graph considered here are simple undirected graph without loops or multiple edges. As usual p,q denote the number of vertices and edges of a graph G respectively. A path on p vertices is denoted by Pp.
A set D ⊆ V of vertices in a graph G is a dominating set if every vertex v in V – D is adjacent to a vertex in D. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by (G).
A Dominating set D of a graph G is a connected dominating set if the induced subgraph < D > is connected. The number of vertices in a minimum connected dominating set is defined as the Connected Domination number of a graph G and it is denoted by (G).
Let G = (V, E) be a simple connected graph with p vertices and q edges . If G1, G2,...,Gn
are connected edge disjoint subgraphs of G with E(G) = E(G1)∪E(G2)∪... E(Gn), then (G1, G2
,...,Gn) is said to be a Decomposition of G.
Definition 1.1 1 : A Decomposition (G1, G2,...,Gn) of G is said to be CDD if (i) E(G) = E(G1) ∪ E(G2) ∪ ...∪ E(Gn) (ii) Each Gi is connected (iii) (Gi) = i, 1 ≤ i ≤ n .
Definition 1.2 1 : A Decomposition (G1, G2,….,Gn) of G is said to be CDPD if (i) admits CDD (ii) q( ) = 1 or 2 (iii) q( ) = + 1, i = 2,3,...,n (iv) Each (1 ≤ ≤ n) is a path.
Remark 1.3 : In CDPD (G1, G2,...,Gn), if q(G1) = 1, then G1 = P2, G2 = P4, G3 = P5,..., Gn = Pn + 2 . If q(G1) = 2, then G1 = P3, G2 = P4, G3 = P5,...,Gn = Pn + 2 .
Theorem 1.4 1 : If a graph G admits CDPD (G1, G2,...,Gn) , then q(G) =
( )
− 1 ( ) = 1, ∀ ∈
( )
( ) = 2, ∀ ∈
2. CONNECTED DOMINATION PATH DECOMPOSITION POLYNOMIAL
Definition 2.1: Let G be a graph which admits CDPD(G1, G2,….,Gn) and let ℳ(G, q(Gt)) be the family of subgraphs (That is, path) with size q(Gt) and m(G, q(Gt)) = | ℳ(G, q(Gt))|. Then the CDPD polynomial of a graph G is defined as
M(G, x) = ∑ ( , ( )) ( ) Example 2.2: Consider a diamond graph G
Here G1 = P3 and G2 = P4
ℳ(G,q(G1)):
ℳ(G,q(G2)):
m(G,q(G1)) = 8, m(G,q(G2)) = 6. M(G, x) = 8x2 +6x3
3. CONNECTED DOMINATION PATH DECOMPOSITION POLYNOMIAL OF PATH The path Pp admits CDPD in two ways.
1. If Pp admits CDPD (P3, P4,...,Pn + 2), then p = , n ∈ 2. If Pp admits CDPD (P2, P4,...,Pn + 3), then p = , n ∈
Theorem 3.1: If Pp admits CDPD (P3, P4,...,Pn+ 2), then M (Pp, x) = ∑ ( − + 1) Proof: Let {u1, u2,..., up} be the vertices of Pp. Assume that G1 = P3 , G2 = P4,..., Gn = Pn + 2.
Then, M (Pp, x) = ∑ ( , ( )) ( )
= m( , ( )) ( ) + m( , ( )) ( ) + .... + m( , ( )) ( ) Let S1, S2 ,…, S p – 2 be the subgraphs of G and each Sk starts with uk, where k = 1, 2,…, p – 2.
If we consider the size of each Sk as 2, then E(S1) = {u1u2, u2u3}, E(S2) = {u2u3, u3u4}, E(S3) = {u3u4, u4u5},…, E(Sp - 2) = { u p - 2u p - 1,u p - 1u p }. Therefore, |ℳ( , ( ))| = p – 2.
That is, m( , ( )) = p – 2
If we consider the size of each Sk as 3, then E(S1) = {u1u2, u2u3, u3u4}, E(S2) = {u2u3, u3u4, u4u5}, E(S3) = {u3u4, u4u5, u5u6},…, E(Sp - 3) = { u p - 3u p - 2, u p - 2u p - 1, u p - 1u p }. Therefore,
|ℳ( , ( ))| = p – 3. That is, m( , ( )) = p – 3
If we consider the size of each Sk as 4 , then E(S1) = {u1u2, u2u3, u3u4, u4u5}, E(S2) = {u2u3, u3u4, u4u5, u5u6}, E(S3) = {u3u4, u4u5, u5u6, u6u7 },…, E(Sp - 4) = { u p - 4u p - 3, u p - 3u p - 2, u p - 2u p - 1, u p - 1u p }. Therefore, |ℳ( , ( ))| = p – 4. That is, m( , ( )) = p – 4.
Continuing in this way, If we consider the size of each Sk as n + 1, then E(S1) = {u1u2, u2u3, u3u4, u4u5, …, u n + 1u n + 2}, E(S2) = {u2u3, u3u4, u4u5, u5u6,…, u n + 2u n + 3 }, E(S3) = {u3u4,u4u5, u5u6, u6u7,…, u n + 3u n + 4 },…, E( ) = { , – ,…,
u p - 1u p }. Therefore, |ℳ( , ( ))| = p – + 1. That is, m( , ( )) = p – + 1
Therefore, M (Pp, x) = (p - 2) x2 + (p - 3) x3 + .... + (p – + 1 ) xn + 1 That is, M (Pp, x) = ∑ ( − + 1)
Illustration 3.2: The number of subgraphs in with size ( ) , t = 1,2,….,n and n = 1,2,...,10 whenever Pp admits CDPD (P3, P4,...,Pn+ 2) is described in the following table.
Table (2) : The number of subgraphs in with size ( ), t = 1,2,….,n and n = 1,2,...,10 whenever Pp admits CDPD (P3, P4,...,Pn + 2)
Remark 3.3: If Pp admits CDPD (P2, P4,..., Pn + 3), then M (Pp, x) =(p - 1) x +
∑ ( − + 2)
4. CONNECTED DOMINATION PATH DECOMPOSITION POLYNOMIAL OF CYCLE
The cycle Cp admits CDPD in two ways.
1. If Cp admits CDPD (P3, P4,...,Pn + 3), then p = + 1, n ∈ 2. If Cp admits CDPD (P2, P4,...,Pn + 3), then p = , n ∈ Theorem 4.1: If Cp admits CDPD (P3, P4,..., Pn+ 3), then M (Cp, x) =∑
Proof: Let {u1, u2,..., up} be the vertices of Cp. Assume that G1 = P3 ,G2 = P4,..., Gn + 1 = Pn + 3
( )
q(Gn) n
n
q(G1) q(G2) q(G3) q(G4) q(G5) q(G6) q(G7) q(G8) q(G9) q(G10)
1 1
2 4 3
3 8 7 6
4 13 12 11 10
5 19 18 17 16 15
6 26 25 24 23 22 21
7 34 33 32 31 30 29 28
8 43 42 41 40 39 38 37 36
9 53 52 51 50 49 48 47 46 45
10 64 63 62 61 60 59 58 57 56 55
= m( , ( )) ( ) + m( , ( )) ( ) + .... + m( , ( )) ( )
Let S1, S2 ,…, Sp be the subgraphs of G and each Sk starts with uk, where k = 1, 2,…, p.
If we consider the size of each Sk as 2, then E(S1) = {u1u2, u2u3}, E(S2) = {u2u3, u3u4}, E(S3) = {u3u4, u4u5},…, E(Sp - 2) = { u p u 1 ,u 1u 2}. Therefore, |ℳ( , ( ))| = p. That is, m( , ( )) = p
If we consider the size of each Sk as 3, then E(S1) = {u1u2, u2u3, u3u4}, E(S2) = {u2u3, u3u4, u4u5}, E(S3) = {u3u4, u4u5, u5u6},…, E(Sp ) = { u p u1, u1u2, u2u 3 } .Therefore,
|ℳ( , ( ))| = p. That is, m( , ( )) = p
If we consider the size of each Sk as 4, then E(S1) = {u1u2, u2u3, u3u4, u4u5}, E(S2) = {u2u3, u3u4, u4u5, u5u6}, E(S3) = {u3u4, u4u5, u5u6, u6u7},…, E(Sp ) = { u p u 1, u 1u 2, u2u 3, u3u4 }.
Therefore, |ℳ( , ( ))| = p. That is, m( , ( )) = p
Continuing in this way, If we consider the size of each Sk as n + 2, then E(S1) = {u1u2, u2u3, u3u4, u4u5, …, u n + 2u n + 3}, E(S2) = {u2u3, u3u4, u4u5, u5u6,…, u n + 3u n + 4 }, E(S3) = {u3u4,u4u5,
u5u6, u6u7,…, u n + 4u n + 5 },…, E(Sp) = { , , u2u3, …, u n + 1u n + 2} .Therefore,
|ℳ( , ( ))| = p . That is, m( , ( )) = p Therefore, M (Cp, x)= p x2 + p x3 + ... + p xn + 2
That is, M (Cp, x) = ∑
Illustration 4.2: The number of subgraphs in Cp with size ( ) , t = 1,2,…,n + 1 and n = 1,2,...,9 whenever Cp admits CDPD (P3, P4,...,Pn+ 2) is described in the following table.
Table (3) : The number of subgraphs in Cp with size ( ) , t = 1,2,…,n + 1 and n = 1,2,...,9 whenever Cp admits CDPD (P3, P4,...,Pn + 2)
q(Gn) n n n
q(G1) q(G2) q(G3) q(G4) q(G5) q(G6) q(G7) q(G8) q(G9) q(G10)
1 5 5
2 9 9 9
3 14 14 14 14
4 20 20 20 20 20
5 27 27 27 27 27 27
6 35 35 35 35 35 35 35
7 44 44 44 44 44 44 44 44
8 54 54 54 54 54 54 54 54 54
9 65 65 65 65 65 65 65 65 65 65
Remark 6.3: If Cp admits CDPD (P2, P4,..., Pn+ 3), then M (Cp, x) = p x + ∑
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