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This paper is motivated by the concept of the signed k-domination problem and dedicated to the complexity of the problem on graphs. For any fixed nonnegative integer k, we show that the signed k-domination problem is NP-complete for doubly chordal graphs. For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination problem can be solved in polynomial time. We also show that the problem is linear-time solvable for trees, interval graphs, and chordal comparability graphs.

Let G = (V, E) be a finite, undirected, simple graph. For any vertex

Before presenting the NP-complete results, we restate the signed k-domination problem as decision problems as follows: Given a graph G = (V, E) and a nonnegative integer k and an integer

Theorem 1 [

Theorem 2. For any fixed integer

Proof. Clearly, the signed k-domination problem on doubly chordal graphs is in NP. By Theorem 1, the signed 0-domination and 1-domination problems on doubly chordal graphs are NP-complete. In the following, we show the NP-completeness of the signed k-domination problem on doubly chordal graphs by a polynomial-time reduction from the signed

Let

1) We construct a new vertex u and connect u to every vertex of G.

2) We construct

Clearly, the graph H is a doubly chordal graph [

Suppose that g is a minimum signed

Conversely, let

Therefore,

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.

Let

For

Farer [

Let

1) If

2)

The L-domination number of G, denoted by

Theorem 4 [

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

Proof. Clearly,

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 [

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 [

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 v_{1},v_{2},…,v_{n}. For any _{i}) < r(v_{j}) and l(v_{j}) < r(v_{i}) if v_{i} is adjacent to v_{j} in G. Therefore, the vertices of N_{i}[v_{i}] form a clique and v_{i} is a simplicial vertex of G_{i}. The ordering v_{1},v_{2},…,v_{n} is a perfect elimination ordering and can be obtained in linear time.

For i < j < k, we assume _{i}[v_{i}] in G_{i}. Since v_{1},v_{2},…,v_{n} is a perfect elimination ordering, v_{j} is adjacent to v_{k} and r(v_{j}) < r(v_{k}) and l(v_{k}) < r(v_{i}) < r(v_{j}). Then, every v_{p} in N_{i}[v_{j}] is adjacent to v_{k}. We have_{1},v_{2},…,v_{n} 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 v_{1},v_{2},…,v_{n} is a simple elimination ordering if for each_{i} is a simple vertex of the subgraph G_{i} induced by the vertices v_{i},v_{2},…,v_{n}.

A simple elimination ordering of a chordal comparability graph can be obtained in linear time [

The 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 [

Theorem 10 [

1) A graph consisting of only one vertex is distance-hereditary, and the twin set is the vertex itself.

2) If

3) If

4) If

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 [

Definition 1. Suppose that

1)

2) The function

3) For a vertex

We define

We give the following lemmas to compute

Lemma 2. Suppose that

Lemma 3. Suppose that

where

Lemma 4. Suppose that

where

Lemma 5. Suppose that

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.

This research was partially supported under Research Grants: NSC-102-2221-E-130-004 and MOST-103-2221- E-130-009 in Taiwan.

Chuan-Min Lee,Cheng-Chien Lo,Rui-Xin Ye,Xun Xu,Xiao-Han Shi,Jia-Ying Li, (2015) Remarks on the Complexity of Signed k-Domination on Graphs. Journal of Applied Mathematics and Physics,03,32-37. doi: 10.4236/jamp.2015.31005