Remarks on the Complexity of Signed k-Domination on Graphs

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.


Introduction
Let G = (V, E) be a finite, undirected, simple graph.For any vertex for any subset S of V and let f(V) be the weight of f.In 2012, Wang [1] studied the notion of signed k-domination on graphs as follows.Let k be a fixed nonnegative integer and let G = (V, E) be a graph.A signed k-dominating function of G is a function : The signed k-domination number of G, denoted by , ( ), k S G γ is the minimum weight of a signed k-dominating function of G.The signed k-domination problem is to find a signed k-dominating function of G of minimum weight.Clearly, the signed k-domination problem is the signed domination problem if k = 1 [2].Wang [1] presented several sharp lower bounds of these numbers for general graphs.In this paper, we study the signed k-domination problem for several well-known classes of graphs such as doubly chordal graphs, strongly chordal graphs, distance-hereditary graphs, trees, interval graphs, and chordal comparability graphs.

NP-completeness Results
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 λ , is , ( ) [4] For any integer k = 0 or 1, the signed k-domination problem on doubly chordal graphs and bipartite planar graphs is NP-complete Theorem 2. For any fixed integer 2, k ≥ the signed k-domination problem on doubly chordal graphs is NP-complete.
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 ( 1) k − -domination problem on doubly chordal graphs.Let ( , ) G V E = be a doubly chordal graph with |V| = n.A clique is a subset of pairwise adjacent vertices in a graph.If a clique consists of j vertices, then it is called a j-clique.We construct a graph H from G by the following steps.1) We construct a new vertex u and connect u to every vertex of G.
and connect the vertex u to every vertex of Clearly, the graph H is a doubly chordal graph [5]- [8] and can be constructed in polynomial time.In the following, we show that , .
It can be easily verified that h is a signed k-dominating function of H.We have , k By the construction of H, the vertex u is adjacent to every vertex v of G.We know that ( [ ]) .

Polynomial-Time Solvable Results
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.

Strongly Chordal Graphs
A perfect elimination order- ing is called a strong elimination ordering if it satisfies the following condition: . 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 ( log ) O m n [11] or 2 ( ) O n 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. }.
satisfying the following two conditions: 1) If ( ) , Theorem 4 [3] For any strongly chordal graph G, the L-domination problem can be solved in ( We show a connection between and the signed problem and a special case of the L-domination problem in Theorem 3.
We assume that f is a minimum L-dominating function of G and each v V ∈ is associated with a label ( ) ( , ).
and f is a signed k-dominating function of G.We have , ( ) ( ).
It can be easily verified that g is an L-dominating function of G.We have , ( ) ( ).
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, i v has at most one neighbor in i G for any {1, 2, , }.
N v forms a clique with at least three vertices and it contradicts the assumption that G is a tree.Therefore, the ordering 1 2 , , , n v v v  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 v 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 j v and k v belong to . The ordering v 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 n t ≤ ≤ 1 , the vertex v 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 [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.

Distance-Hereditary Graphs
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 [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.G by a true twin operation.

4) If 1
G and 2 G are disjoint distance-hereditary graphs with the twin sets

TS G
G is said to be obtained from 1 G and 2 G 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 ( , ) G V E = is a distance-hereditary graph.Let ( ) TS G be the twin set of .
G  ( , ) G V E = by a false twin operation.Then, also use V(G) and E(G) to denote vertex set and edge set of G, respectively.If nothing else is stated, it is understood that |V(G)| = n and |E(G)| = m.Let Y be a subset of real numbers.Let : f V Y → be a function which assigns to each v V ∈ a value in Y.
Following the discussion above, we know that , domination number of G, denoted by ( ), L G γ 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.
respec- tively, then the graph G obtained by connecting every vertex of 1 ( ) TS G to all vertices of 2 ( ) TS G is a distance-hereditary graph and the twin set of G is 1 ( ).

=
Let a and b be integers such that 0 , a, b)-function : { 1,1} f V → − of G is a function satisfying the following three conditions.1) | ( ) | .a b TS G + = 2) The function f assigns the value 1 to a vertices in TS(G) and the value 1 − to b vertices in TS(G).We define ( , , , ) max{ ( ( )) | is a ( , , )-function of }.If there does not exist a (t, a, b)function of G,then ( , , , ).We give the following lemmas to compute ( , , , ) G t a b γ 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 [

Lemma 3 .
Suppose that ( , ) G V E= is formed from two disjoint distance-hereditary graphs 1