ISRN Combinatorics

Volume 2013 (2013), Article ID 146595, 3 pages

http://dx.doi.org/10.1155/2013/146595

Research Article

On the Domination Polynomial of Some Graph Operations

Saeid Alikhani1,2

¹Department of Mathematics, Yazd University, Yazd 89195-741, Iran

²School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Received 9 June 2013; Accepted 28 July 2013

Academic Editors: L. Feng and J. Siemons

Copyright © 2013 Saeid Alikhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . Every root of is called the domination root of . In this paper, we study the domination polynomial of some graph operations.

1. Introduction

Let be a simple graph. For any vertex , the open neighborhood of is the set and the closed neighborhood is the set . For a set , the open neighborhood of is and the closed neighborhood of is . A set is a dominating set if , or equivalently, every vertex in is adjacent to at least one vertex in . An -subset of is a subset of of cardinality . Let be the family of dominating sets of which are -subsets and let . The polynomial is defined as domination polynomial of [1]. This polynomial has been introduced by the author in his Ph.D. thesis in 2009 [2]. A root of is called a domination root of . More recently, domination polynomial has found application in network reliability [3]. For more information and motivation of domination polynomial and domination roots refer to [1, 2].

The join of two graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and . The corona of two graphs and , is the graph formed from one copy of and copies of , where the th vertex of is adjacent to every vertex in the th copy of [4]. the Cartesian product of two graphs and is denoted by , is the graph with vertex set and edges between two vertices and if and only if either and or and .

In this paper, we study the domination polynomials of some graph operations.

2. Main Results

As is the case with other graph polynomials, such as chromatic polynomials and independence polynomials, it is natural to consider the domination polynomial of composition of two graphs. It is not hard to see that the formula for domination polynomial of join of two graphs is obtained as follows.

Theorem 1 (see [1]). Let and be graphs of orders and , respectively. Then

It is obvious that this operation of graphs is commutative. Using this product, one is able to construct a connected graph with the number of dominating sets , where is an arbitrary odd natural number; see [5].

Let to consider the corona of two graphs. The following theorem gives us the domination polynomial of graphs of the form which is the first result for domination polynomial of specific corona of two graphs.

Theorem 2 (see [1]). Let be a graph. Then if and only if for some graph of order .

It is easy to see that the corona operation of two graphs does not have the commutative property. The following theorem gives us the domination polynomial of .

Theorem 3. For every graph of order , .

Proof. In each graph of the form , we have two cases for a dominating set .

Case  1. includes (the vertex originally in ) and an arbitrary subset of the vertices from the copy of . The generating function for the number of dominating sets of graph in this case is .

Case  2. does not include and it is exactly a dominating set of . In this case is the generating function.

By addition principle, we have .

The following theorem gives a formula for domination polynomial of corona products of two graphs.

Theorem 4. Let and be nonempty graphs of order and , respectively. Then

Proof. By Theorem 3, it suffices to prove that . In the corona of two graphs and , every vertex of is adjacent to all vertices of the corresponding copy of . So, we can delete all edges in in the corona. Therefore, the arising graph is the disjoint union of copies of the corona . Therefore, .

As a consequence of the above theorem, we have the following corollary.

Corollary 5. (i) Let be a connected graph of order . Then, , for some graph , if and only if (see [1]) .

(ii) Let be a graph of order and . Then .

It is interesting that for the classification of graphs with exactly two, three, and four domination roots, we must consider some kinds of corona of two graphs. For more information, see [1].

To study more we need the following theorem.

Theorem 6 (see [2]). If    has   connected components , then .

Now we will consider the Cartesian product of two graphs. First we prove the following easy result.

Theorem 7. If    has    connected components , then

Proof. Since we have , we have the result by Theorem 6.

Despite the above property, it is difficult to determine the domination polynomial of this product, even in such simple cases as the grid graphs .

Now we consider another operation of two graphs. Let and be graphs, with . The graph formed by substituting a copy of for every vertex of is formally defined by taking a disjoint copy of , , for every vertex of and joining every vertex in to every vertex in if and only if is adjacent to in .

The following result is also proven in [6, Lemma 3].

Theorem 8. For any graph , .

Proof. Note that the closed neighborhood of the vertex of graph is . To make a dominating set of , suppose that is a dominating set for . It is easy to see that is a dominating set of , where is a family of arbitrary nonempty subsets of . Therefore, every corresponds to all nonempty subsets of which have the generating function . So we have the result.

We would like to obtain some corollaries. We recall the following theorems.

Theorem 9 (see [7]). (i) For every ,

with the initial values .

(ii) For every ,

with the initial values .

Here we consider the graphs obtained by selecting one vertex in each of triangles and identifying them. Some call them Dutch Windmill graphs [8, 9] and friendship graphs.

Theorem 10. For every ,

Proof. It is easy to see that is join of and . Now by Theorem 1, we have

Theorem 8 can be used to generalize recurrence relations for the domination polynomial of some families of graphs. For example, we state and prove the following theorem.

Theorem 11. (i) Suppose that . Then

(ii) Suppose that . Then

(iii)

Proof. (i) From Theorem 8, we have . Now by Part (i) of Theorem 9, we have the result.

(ii) From Theorem 8, we have . Now by Part (ii) of Theorem 9, we have the result.

(iii) From Theorem 8, we have . Now by Theorem 10, we have the result.

3. Conclusion

In this paper, we studied the domination polynomials of some graph operations. There are some open problems which are interesting to consider.

(i) What is the basic formula for the domination polynomial of the Cartesian product of two graphs?

For two graphs and , let be the graph with vertex set and such that vertex is adjacent to vertex if and only if is adjacent to (in ) or and is adjacent to (in ). The graph is the lexicographic product (or composition) of and and can be thought of as the graph arising from and by substituting a copy of for every vertex of . There is a main problem.

(ii) How can compute the domination polynomial of Lexicographic product of the two graphs?

Acknowledgments

The author would like to express his gratitude to the referees for his comments. The research was in part supported by a Grant from IPM (no. 91050015).

References

S. Akbari, S. Alikhani, and Y.-h. Peng, “Characterization of graphs using domination polynomials,” European Journal of Combinatorics, vol. 31, no. 7, pp. 1714–1724, 2010.

S. Alikhani, Dominating sets and domination polynomials of graphs [Ph.D. thesis], Universiti Putra Malaysia, 2009.

K. Dohmen and P. Tittmann, “Domination reliability,” Electronic Journal of Combinatorics, vol. 19, no. 1, paper 15, 2012.

R. Frucht and F. Harary, “On the corona of two graphs,” Aequationes Mathematicae, vol. 4, pp. 322–325, 1970.

S. Alikhani, “The domination polynomial of a graph at -1,” Graphs and Combinatorics, 2012.

J. Brown and J. Tufts, “On the roots of domination polynomials,” Graphs and Combinatorics, 2013.

S. Alikhani and Y.-H. Peng, “Dominating sets and domination polynomials of certain graphs. II,” Opuscula Mathematica, vol. 30, no. 1, pp. 37–51, 2010.

S. Alikhani and E. Deutsch, “Graphs with domination roots in the right half-plane,” In press

http://mathworld.wolfram.com/DutchWindmillGraph.html