In this section, we show that the signed k-domination problem is polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs and linear-time solvable for trees, interval graphs, and chordal comparability graphs.
3.1. Strongly Chordal GraphsLet be a graph. A clique is a subset of pairwise adjacent vertices of A vertex is simplicial if and only if all vertices of form a clique. The ordering of the vertices of is a perfect elimination ordering of G if for all, is a simplicial vertex of the subgraph of G induced by [9]. Let denote the closed neighborhood of in. A perfect elimination ordering is called a strong elimination ordering if it satisfies the following condition:
For if and belong to in, then.
Farer [10] showed that a graph is strongly chordal if and only if it has a strong elimination ordering. Currently, the fastest algorithm to recognize a strongly chordal graph and give a strong elimination ordering takes [11] or time [12]. Strongly chordal graphs include many interesting classes of graphs such as trees, block graphs, interval graphs, and directed path graphs [13]. In the paper [3], Lee and Chang introduced the concept of L-domination. The definition of L-domination is as follows.
Let be fixed integer such that and Let Y be the set Suppose that is a graph. Let L be a labeling function which assigns to each a label where and is a fixed integer. An L-dominating function of a graph is a function satisfying the following two conditions:
1) If then.
2) for every vertex
The L-domination number of G, denoted by is the minimum weight of an L-dominating function of G. The L-domination domination problem is to find an L-dominating function of G of minimum weight. Lee and Chang obtained the following result.
Theorem 4 [3] For any strongly chordal graph G, the L-domination problem can be solved in time if a strong elimination ordering of G is given.
We show a connection between and the signed k-domination problem and a special case of the L-domination problem in Theorem 3.
Theorem 5. Suppose that and Let k be a nonnegative integer and let be a graph in which each is associated with a label. Then, a minimum L-dominating function of G is equivalent to a minimum signed k-dominating function of G.
Proof. Clearly, We assume that is a minimum L-dominating function of G and each is associated with a label Then, and is a signed k-dominating function of G. We have Conversely, we assume that is a minimum signed k-dominating set of G. Then, for every vertex It can be easily verified that g is an L-dominating function of G. We have Following the discussion above, we know that and Hence, and the theorem holds.
Theorem 6. For any nonnegative integer k, the signed k-domination problem on a strongly chordal graph G can be solved in O(n + m) time if a strong elimination ordering of G is given.
Proof. The theorem follows from Theorems 4 and 5.
Theorem 7. For any nonnegative integer k, the signed k-domination problem is linear-time solvable for trees.
Proof. Trees are both chordal and strongly chordal [13]. Let G be a tree. A perfect elimination ordering of the vertices in G can be obtained in linear time [14]. Since G is a tree, has at most one neighbor in for any Otherwise, forms a clique with at least three vertices and it contradicts the assumption that G is a tree. Therefore, the ordering is also a strong elimination ordering of G. Following Theorem 6, we know that the signed k-domination problem is linear-time solvable for trees.
Theorem 8. For any nonnegative integer k, the signed k-domination problem is linear-time solvable for interval graphs.
Proof. An interval graph G is the intersection graph of a set of intervals on a line. That is, each interval corresponds to a vertex of G and two vertices are adjacent if and only if the corresponding intervals intersect. The set of intervals constitutes an interval model of the graph. Booth and Lueker [15] gave the first linear-time algorithm for recognizing interval graphs and constructing interval models for the interval graphs.
Let I be an interval model of an interval graph G. Each interval in the interval model has a right endpoint and a left endpoint. Without loss of generality, we may assume that all endpoints of the intervals in I are pairwise distinct, since, when they are not, it is easy to make this true without altering the represented graph. Let l(v) and r(v) denote the left and right endpoints of the interval corresponding to v. We order the vertices of G by the increasing order of right endpoints of the intervals in I, and let the ordering be v1,v2,…,vn. For any we know that r(vi) < r(vj) and l(vj) < r(vi) if vi is adjacent to vj in G. Therefore, the vertices of Ni[vi] form a clique and vi is a simplicial vertex of Gi. The ordering v1,v2,…,vn is a perfect elimination ordering and can be obtained in linear time.
For i < j < k, we assume and belong to Ni[vi] in Gi. Since v1,v2,…,vn is a perfect elimination ordering, vj is adjacent to vk and r(vj) < r(vk) and l(vk) < r(vi) < r(vj). Then, every vp in Ni[vj] is adjacent to vk. We have. The ordering v1,v2,…,vn is also a strong elimination ordering of G. By Theorem 6, we know that the signed k-domination problem is linear-time solvable for interval graphs.
Theorem 9. For any nonnegative integer k, the signed k-domination problem is linear-time solvable for chordal comparability graphs.
Proof. Let G = (V, E) be a graph. A vertex v in G is a simple vertex if for any two neighbors x and y of v, either the closed neighborhood of y is a subset of the closed neighborhood of x or the closed neighborhood of x is a subset of the closed neighborhood of y. An ordering v1,v2,…,vn is a simple elimination ordering if for each, the vertex vi is a simple vertex of the subgraph Gi induced by the vertices vi,v2,…,vn.
A simple elimination ordering of a chordal comparability graph can be obtained in linear time [16]. Sawada and Spinrad [17] presented a linear-time algorithm to transform a simple elimination ordering of a strongly chordal graph to a strong elimination ordering. Therefore, the theorem is true.
3.2. Distance-Hereditary GraphsThe distance between two vertices u and v of a graph G is the number of edges of a shortest path from u to v. If any two distinct vertices have the same distance in every connected induced subgraph containing them, then G is a distance-hereditary graph. In 1997, Chang, Hsieh, and Chen [18] showed that distance-hereditary graphs can be defined recursively.
Theorem 10 [18] Distance-hereditary graphs can be defined as follows.
1) A graph consisting of only one vertex is distance-hereditary, and the twin set is the vertex itself.
2) If and are disjoint distance-hereditary graphs with the twin sets and, respectively, then the graph is a distance-hereditary graph and the twin set of is G is said to be obtained from and by a false twin operation.
3) If and are disjoint distance-hereditary graphs with the twin sets and, respectively, then the graph obtained by connecting every vertex of to all vertices of is a distance-hereditary graph and the twin set of is G is said to be obtained from and by a true twin operation.
4) If and are disjoint distance-hereditary graphs with the twin sets and, respectively, then the graph obtained by connecting every vertex of to all vertices of is a distance-hereditary graph and the twin set of is G is said to be obtained from and by a pendant vertex operation.
Following Theorem 10, a distance-hereditary graph G can be represented as a binary ordered decomposition tree and the decomposition tree can be obtained in linear-time [18]. In this decomposition tree, each leaf is a single vertex graph, and each internal node represents one of the three operations: pendant vertex operation (labeled by P), true twin operation (labeled by T), and false twin operation (labeled by F). Therefore, the decomposition tree is called a PTF-tree.
Definition 1. Suppose that is a distance-hereditary graph. Let be the twin set of Let a and b be integers such that and A (t, a, b)-function of G is a function satisfying the following three conditions.
1)
2) The function assigns the value 1 to a vertices in TS(G) and the value to b vertices in TS(G).
3) For a vertex if Otherwise,
We define If there does not exist a (t, a, b)- function of G, then It is clear that
![]()
We give the following lemmas to compute for a distance-hereditary graph G. The correctness of Lemmas 2 - 5 can be proved by the arguments similar to those for proving Lemmas 1 - 4 in Section III. B of the paper [4].
Lemma 2. Suppose that is a graph of only one vertex v. Then,
![]()
Lemma 3. Suppose that is formed from two disjoint distance-hereditary graphs and by a false twin operation. Then,
,
where and
Lemma 4. Suppose that is formed from two disjoint distance-hereditary graphs and by a true twin operation. Then,
,
where and
Lemma 5. Suppose that is formed from two disjoint distance-hereditary graphs and by a pendant vertex operation. Then,
,
where
Theorem 11. For any nonnegative integer k, the signed k-domination problem can be solved in polynomial time for distance-hereditary graphs.
Proof. Following Lemmas 2 - 5 and the recursive definition of distance-hereditary graphs in Theorem 10, we can design a dynamic programming algorithm to compute the signed k-domination number of a distance-here- ditary graph G in polynomial time. Moreover, it is not difficult to see that a minimum signed k-dominating function of a distance-hereditary graph G can be obtained in polynomial time, too.