Domination polynomials of k-tree related graphs

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $\gamma(G)$ 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 kind of graphs, whose domination polynomial have no nonzero real roots.


Introduction
Throughout this paper we will consider only simple graphs. Let G = (V, E) be a simple graph.
be the degrees of the vertices v i of a graph in any order. The sequence {d i } n 1 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 K n , P n , C n and S 1,n−1 for a clique, a path, a cycle and a star, all of order n, respectively.
A set S ⊆ V (G) is a dominating set if N [S] = V or equivalently, every vertex in V (G)\S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. A dominating set with cardinality γ(G) is called a γ-set. For a detailed treatment of these parameters, the reader is referred to [13]. Let D(G, i) be the family of dominating sets of a graph G with cardinality i and let d( [2,7]). Thus D(G, x) is the generating polynomial for the number of dominating sets of G of each cardinality. A root of D(G, x) is called a domination root of G.
In [14] it is shown that computing the domination polynomial D(G, x) of a graph G is NP-hard and some examples for graphs for which D(G, x) can be computed efficiently are given. The vertex contraction G/u of a graph G by a vertex u is the operation under which all vertices in N (u) are joined to each other and then u is deleted (see [17]). The following theorem is useful for finding the recurrence relations for the domination polynomials of arbitrary graphs. Theorem 1. [5,14] Let G be a graph. For any vertex u in G we have where p u (G, x) is the polynomial counting the dominating sets of G − u which do not contain any vertex of N (u) in G.
Using Theorem 1 we are able to obtain an easier formula for a graph with at lease one vertex of degree 1. Since every tree has at least two vertices of degree 1, so we can use the following recurrence to obtain the domination polynomials of trees. Corollary 1. [8,14] Let G = (V, E) be a graph, v be a vertex of degree 1 in G and let u be its neighbor. Then If G 1 and G 2 are disjoint graphs of orders n 1 and n 2 respectively, then D(G 1 ∪ G 2 , x) = D(G 1 , x)D(G 2 , x) and where G 1 + G 2 is the join of G 1 and G 2 , formed from G 1 ∪ G 2 by adding in all edges between a vertex of G 1 and a vertex of G 2 (see [2]).
The domination polynomials of trees, aside from path graph have not been studied and there is no study for coefficients of D(T, x) for trees T with n vertices. k-trees are generalization of tree which consider in this paper. Actually similar to [16], in this paper we consider k-tree related graphs and study their domination polynomials. Study of the roots of domination polynomials is a interesting ( [1,10]). One of the problem in domination roots is classification and finding graphs with no nonzero real roots. In this paper we present some families related to k-trees which have this property.
In Section 2, we study the domination polynomials for some k-tree related graphs. In Section 3, we present some families of these kind of graphs whose domination polynomials have no nonzero real roots.

Domination polynomials of k-tree related graphs
In this section we study the domination polynomials for some k-tree related graphs. The class of k-trees is a very important subclass of triangulated graphs. Harary and Palmer [12] first introduced 2-trees in 1968. Beineke and Pippert [9] gave the definition of a k-tree in 1969.
In the literature on k-trees, there are interesting applications to the study of computational complexity.
Definition 1. For a positive integer k, a k-tree, denoted by T k n , is defined recursively as follows: The smallest k-tree is the k-clique K k . If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the larger graph is a k-tree with n + 1 vertices.
An independent set in a graph G is a set of pairwise non-adjacent vertices.
Definition 2. Let K k be a k-clique and S be an independent set of n − k vertices. A (k, n)-star, denoted by S k,n−k , is defined as S k,n−k = K k + S. Figure 1). A helpful characteristic of the (k, n)-path P k n is that we may order the vertices such a vertex ordering is referred to as a presentation.
. . , v n } defined as above and an edge joining v 1 to v n , where n ≥ k + 2.
Definition 5. If G is a (k, n)-cycle of order n and v is a vertex not in G, then G + v is called a (k, n)-wheel and denoted by W k n .
Notice that P 1 n , C 1 n , W 1 n and S 1,n−1 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 K 1,n−1 , we have D(K 1,n , x) = x(1 + x) n−1 + x n−1 . The following are recurrences for the domination polynomials of paths and cycles ( [6]).
(iii) Since the (k, n)-wheel W k n , has a vertex v of degree n − 1, so (iii) holds.
(iv) Since the k-star graph S k,n−k , has k vertices of degree n − 1, so (iv) holds.
The following theorem gives a recurrence formula for the domination polynomial of (k, n)-path graphs.
Proof. If n ≤ k + 1, then P k n ∼ = K n . For every k + 2 ≤ n, we use Theorem 1 for the last vertex of P k n and since (by the definition of P k n ) the first k + 1 and the last k + 1 vertices form two cliques, Obviously p u (P k n , x) is the polynomial counting the dominating sets of P k n−k−1 contains the vertex v n−k−1 , but finding this polynomial involve complex calculations. We brought this polynomial for n ≤ 2k + 6 in this theorem. Therefore we have the result.
In general, finding the domination polynomial of a graph is a very difficult problem. In [14] 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 (k, n)cycle graphs, If n ≤ k + 2, then C k n ∼ = K n . Consequently in this case D(C k n , x) = D(K n , x). For every k + 3 ≤ n, until now all attempts to find formulas for D(C k n , x) failed.
The following theorem gives a formula for the domination polynomial of (k, n)-wheel graphs.
Theorem 5. For a (k, n)-wheel W k n , we have Proof. Since W k n = C k n + K 1 , then The following theorem gives a formula for the domination polynomial of k-star graphs, which is concluded of the fact, k-star graph is the join of complete graph K k and independent set S (empty graph O n−k ).
It suffices to show that every dominating set of size j = 1, . . . , n is accounted for exactly once in the above statement. Clearly every non-empty subset of {v 1 , v 2 , · · · , v k } is dominating set of k-star graphs. These sets can be extended with any number of vertices {v k+1 , · · · , v n }. Also obviously the set {v k+1 , · · · , v n } is a dominating set of k-star graphs. It is easy to see that there is no another method to make a dominating set for k-star graphs. Therefore we have the result.
The value of a graph polynomial at a specific point can give sometimes a surprising information about the structure of the graph [4,15]. The following simple results give the domination polynomial of k-tree related graphs at −1.

Some families of graphs with no nonzero real domination roots
In [1] authors asked that which graphs have no nonzero real domination roots?
In this section we would like to obtain more results related to this problem. We need some preliminaries.
For two graphs G = (V, E) and H = (W, F ), the corona G • H is the graph arising from the disjoint union of G with |V | copies of H, by adding edges between the ith vertex of G and all vertices of ith copy of H [11]. It is easy to see that the corona operation of two graphs does not have the commutative property.
We need the following theorem which is for computation of domination polynomial of corona products of two graphs.
Theorem 7. [3,14] Let G = (V, E) and H = (W, F ) be nonempty graphs of order n and m, respectively. Then Here we will discuss roots of domination polynomial of k-star graphs.
(i) For odd natural n and even natural k, no nonzero real numbers is domination root of S k,n−k .
(ii) For even natural n and even natural k, there is exactly one nonzero real domination root of S k,n−k .
Proof. By Theorem 6, for every n > k, D(S k,n−k , x) = (1 + x) n−k ((1 + x) k − 1) + x n−k . If D(S k,n−k , x) = 0, then we have Now we are ready to prove two cases of this theorem: (i) First suppose that x ≥ 0. Obviously the above equality is true just for real number 0, since for nonzero real number the left side of equality is positive but the right side is negative. Now suppose that x < −1. In this case the left side is negative and the right side −x n−k is greater than +1, a contradiction. Finally we shall consider −1 < x < 0. This case is similar to the second case when we substitute x with 1 x .
(ii) First suppose that x ≥ 0 and x < −1. Obviously the above equality is true just for real number 0, since for nonzero real number the left side of equality is positive but the right side is negative. Now suppose that −1 < x < 0. This equation has only one real root in (−1, 0).
Remark. Using Maple we have shown the domination roots of S 4,n−4 for 5 ≤ n ≤ 44 in Figure   2. Here we construct a sequence of graphs, which their domination roots are the same as the domination roots of the k + 1-star graphs. Therefore we have the result.