Abstract
Let G be a simple graph of order n. The domination polynomial of G is the polynomial , where d(G, i) is the number of dominating sets of G of size i and γ(G) is the domination number of G. In this paper, we study the domination polynomials of several classes of k-tree related graphs. Also, we present families of these kinds of graphs, whose domination polynomials have no nonzero real roots.
1. Introduction
Throughout this paper, we will consider only simple graphs. Let be a simple graph. For we use for the subgraph induced by . For any vertex , the open neighborhood of is the set and the closed neighborhood of is the set . For a set , the open neighborhood of is and the closed neighborhood of is . For every vertex , the degree of is the number of edges incident with and is denoted by . Let , be the degrees of the vertices of a graph in any order. The sequence is called the degree sequence of the graph. A clique in a graph is a subset of its vertices such that every two vertices in the subset are connected by an edge. We use , , , and for a clique, a path, a cycle, and a star, all of order , respectively.
A set is a dominating set if or, equivalently, every vertex in is adjacent to at least one vertex in . The domination number is the minimum cardinality of a dominating set in . A dominating set with cardinality is called a -set. For a detailed treatment of these parameters, the reader is referred to [1]. Let be the family of dominating sets of a graph with cardinality and let . The domination polynomial of is defined as , where is the domination number of (see [2, 3]). Thus, is the generating polynomial for the number of dominating sets of of any cardinality. A root of is called a domination root of .
In [4], it is shown that computing the domination polynomial of a graph is NP-hard and some examples for graphs for which can be computed efficiently are given. The vertex contraction of a graph by a vertex is the operation under which all vertices in are pairwise joined to each other and then is deleted (see [5]). The following theorem is useful for finding the recurrence relations for the domination polynomials of arbitrary graphs.
Theorem 1 (see [4, 6]). Let be a graph. For any vertex in , one has where is the polynomial counting the dominating sets of which do not contain any vertex of in .
Using Theorem 1 we are able to obtain an easier formula for a graph with at least one vertex of degree 1. Since every tree has at least two vertices of degree 1, we can use the following recurrence to obtain the domination polynomials of trees.
Corollary 2 (see [4]). Let be a graph, let be a vertex of degree in , and let be its neighbor. Then,
If and are disjoint graphs of orders and , respectively, then and where is the join of and , formed from by adding in all edges between a vertex of and a vertex of (see [2]).
The domination polynomials of trees, aside from path graph, have not been studied and there is no study for coefficients of for trees with vertices. In this paper, -trees, a generalization of trees, are considered. We study, similar to [7], -tree related graphs and study their domination polynomials. An interesting study of the roots of domination polynomials is done in [8, 9]. Classification of the roots of domination polynomials is difficult to do, as well as finding graphs with no nonzero real roots. In this paper, we present -tree related families which have this prementioned property.
In Section 2, we study the domination polynomials for some -tree related graphs. In Section 3, we present some families of these kinds of graphs whose domination polynomials have no nonzero real roots.
2. Domination Polynomials of k-Tree Related Graphs
In this section, we study the domination polynomials for some -tree related graphs. The class of -trees is a very important subclass of triangulated graphs. Harary and Palmer [10] first introduced -trees in 1968. Beineke and Pippert [11] gave the definition of a -tree in 1969. In the literature on -trees, there are interesting applications to the study of computational complexity.
Definition 3. For a positive integer , a -tree, denoted by , is defined recursively as follows. The smallest -tree is the -clique . If is a -tree with vertices and a new vertex of degree is added and joined to the vertices of a -clique in , then the larger graph is a -tree with vertices.
An independent set in a graph is a set of pairwise nonadjacent vertices.
Definition 4. Let be a -clique and let be an independent set of vertices. A -star, denoted by , is defined as .
Definition 5. A -path, denoted by , begins with -clique on . For to , let vertex be adjacent to vertices only (see Figure 1).
A helpful characteristic of the -path is that we may order the vertices such that is a -path on vertices for ; such a vertex ordering is referred to as a presentation.
Definition 6. A -cycle, denoted by , consists of a -path on defined as above and an edge joining to , where .
Definition 7. If is a -cycle of order and is a vertex not in , then is called a -wheel and is denoted by .
Notice that , , , and are just the standard path, cycle, wheel, and star, respectively. It follows easily from the domination polynomial of join of two graphs that, for the star graph , we have . The following are recurrences for the domination polynomials of paths and cycles [12].
Theorem 8. For the natural number , (i), where , , and ;(ii), where , , and .
Note that both -cycles and -wheels are not -trees. But they are closely related to -trees. We begin with a simple lemma which was proven in [7] as Proposition 2.
Lemma 9. For any -tree , .
The independence number is the size of a maximum independent set in a graph and is denoted by . The following lemma gives independence numbers for -tree related graphs.
Lemma 10 (see [7]). For each natural number , one has (i),(ii), (iii),(iv).
Now, we present the following domination numbers for -tree related graphs.
Theorem 11. For each natural number , one has (i),(ii),(iii),(iv).
Proof. (i) Since , by the definition of , the degree sequence in this graph is
we have for and for . Thus, (i) holds for . Now, assume . We use induction on . Since any -set of contains only one vertex of the and is a -path with vertices, by induction, . Hence, (i) holds.
(ii) Since , by the definition of , the degree sequence in this graph is
we have for and for . Thus, (ii) holds for . Now, assume and use induction on . Since any -set of contains only one vertex of the and is a -path with vertices, by induction, . Hence, (ii) holds.
(iii) Since the -wheel has a vertex of degree , (iii) holds.
(iv) Since the -star graph has vertices of degree , (iv) holds.
The following theorem gives a recurrence formula for the domination polynomial of -path graphs.
Theorem 12. If , then . For every , where
Proof. If , then . For every , we use Theorem 1 for the last vertex of and since (by the definition of ) the first and the last vertices form two cliques, we have . It is clear that . Obviously, is the polynomial counting the dominating sets of and contains the vertex . However, finding this polynomial involves complex calculations. We therefor give this polynomial only for . Therefore, we have the result.
In general, finding the domination polynomial of a graph is a very difficult problem. In [4], Kotek et al. showed that there exist recurrence relations for the domination polynomial which allow for efficient schemes to compute the polynomial for some types of graphs. Consider the -cycle graphs. If , then . Consequently, in this case, . Until now, for every , all attempts to find formulas for have failed.
The following theorem gives a formula for the domination polynomial of -wheel graphs.
Theorem 13. For a -wheel , one has
Proof. Since , then
The following theorem gives a formula for the domination polynomial of -star graphs, which is derived from the fact that the -star graph is the join of complete graph and independent set (empty graph ).
Theorem 14. For every and ,
Proof. Let be the -star graph with vertex set . It suffices to show that every dominating set of size is accounted for exactly once in the above statement. Clearly, every nonempty subset of is a dominating set of -star graphs. These sets can be extended with any number of vertices . Also, obviously, the set is a dominating set of -star graphs. It is easy to see that there is no other method to form a dominating set for -star graphs. Therefore, we have the result.
Remark 15. It is easy to see that another approach for proving Theorem 14 is using the formula for the domination polynomial of join of two graphs.
The value of a graph polynomial at a specific point can give sometimes a surprising information about the structure of the graph [3, 13]. The following simple results give the domination polynomial of -tree related graphs at .
Corollary 16. For each natural number , the following hold: (i),(ii),(iii).
Proof. (i) Using the domination polynomial of a -path in Theorem 12, for , yields . For every , . We use induction on . Suppose that the statement is true for every -path with vertices; by induction and Lemma 10,
Hence, (i) holds.
(ii) Follows from Theorem 13.
(iii) Follows from Theorem 14 and Lemma 10.
3. Some Families of Graphs with No Nonzero Real Domination Roots
In [8], the authors asked the question: “Which graphs have no nonzero real domination roots?”
In this section, we obtain more results related to this question. We need some preliminaries.
For two graphs and , the corona is the graph arising from the disjoint union of with copies of , by adding edges between the th vertex of and all vertices of th copy of [14]. It is easy to see that the corona operation of two graphs does not have the commutative property.
We need the following theorem which will be used to find the domination polynomial of the corona products of two graphs.
Theorem 17 (see [4, 15]). Let and be nonempty graphs of orders and , respectively. Then,
A -star, , has vertex set where and for .
Now, we will discuss the roots of the domination polynomial of the -star graphs.
Theorem 18. (i)For odd natural and even natural , no nonzero real number is the domination root of .(ii)For even natural and even natural , there is at least one nonzero real domination root of .
Proof. By Theorem 14, for every , . If , then we have Now we prove the two cases of this theorem. (i)First, suppose that . Obviously, equality (13) is true just for , since for nonzero real number the left side of equality is positive but the right side is negative. If , then the left side of equality is negative but the right side is positive. Now, suppose that . In this case, the left side is greater than and the right side is less than , a contradiction. Therefore, in any cases, we have the result.(ii)Let . Similar to Case (i), equality (13) is true only for , since for nonzero real number the left side of equality is positive but the right side is negative. Now, suppose that . In this case, the right side is less than and the left side is greater than , a contradiction. It remains to consider . It is easy to see that , and . So has at least one real root in .
Remark. Using Maple, we have shown the domination roots of for in Figure 2.
Here, we construct a sequence of graphs, of which the domination roots are the same as the domination roots of the -star graphs.
Theorem 19. The domination roots of every graph in the family have the same behavior as the domination roots of -star graphs.
Proof. By Theorem 17, we can deduce that, for each arbitrary graph , Therefore, we have the result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to express their gratitude to the referees for their careful reading and helpful comments.