Algorithmic Aspects of 2-Secure Domination in Graphs

02/05/2020 ∙ by J. Pavan Kumar, et al. ∙ 0

Let G(V,E) be a simple, undirected and connected graph. A dominating set S ⊆ V(G) is called a 2-secure dominating set (2-SDS) in G, if for every pair of distinct vertices u_1,u_2 ∈ V(G) there exists a pair of distinct vertices v_1,v_2 ∈ S such that v_1 ∈ N[u_1], v_2 ∈ N[u_2] and (S ∖{v_1,v_2}) ∪{u_1,u_2 } is a dominating set in G. The 2-secure domination number denoted by γ_2s(G), equals the minimum cardinality of a 2-SDS in G. Given a graph G and a positive integer k, the 2-Secure Domination (2-SDM) problem is to check whether G has a 2-secure dominating set of size at most k. It is known that 2-SDM is NP-complete for bipartite graphs. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We strengthen the NP-complete result for bipartite graphs, by proving this problem is NP-complete for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite graphs. We prove that 2-SDM is linear time solvable for bounded tree-width graphs. We also show that the 2-SDM is W[2]-hard even for split graphs. The Minimum 2-Secure Dominating Set (M2SDS) problem is to find a 2-secure dominating set of minimum size in the input graph. We propose a Δ(G)+1- approximation algorithm for M2SDS, where Δ(G) is the maximum degree of the input graph G and prove that M2SDS cannot be approximated within (1 - ϵ) ln(| V | ) for any ϵ > 0 unless NP ⊆ DTIME(| V |^ O(loglog | V | )). bipartite graphs. A secure dominating set of a graph defends one attack at any vertex of the graph. Finally, we show that the M2SDS is APX-complete for graphs with Δ(G)=4.

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1 Introduction

Let be a simple, undirected and connected graph. For graph theoretic terminology we refer to [19]. For a vertex , the open neighborhood of in is = {}, the closed neighborhood of is defined as . If , then the open neighborhood of is the set . The closed neighborhood of is . Let . Then a vertex is called a private neighbor of with respect to if If further , then is called an external private neighbor (epn) of .

A subset of is a dominating set in if for every , there exists such that . The domination number of is the minimum cardinality of a dominating set in and is denoted by . A set is a -dominating set if every vertex in has at least neighbors in . A dominating set is said to be a secure dominating set (SDS) in if for every there exists such that and is a dominating set of . In this context, we say -defends or is -defended by . The minimum cardinality of a SDS in is called the secure domination number of and is denoted by . Suppose a guard at a vertex of the graph can deal with a problem in its closed neighborhood. A dominating set is said to be secure dominating set if an attack occurs at any vertex of the graph can be -defended by some vertex in its closed neighborhood. However, suppose if two attacks simultaneously happen at any two vertices of the graph, then how to defend both the vertices is an interesting problem. A dominating set is called a -secure dominating set (-SDS) in , if for every pair of distinct vertices there exists a pair of distinct vertices such that , and is a dominating set in . The -secure domination number denoted by , equals the minimum cardinality of a -SDS in and any minimum -secure dominating set is referred as -set of . Given a graph and a positive integer the -Secure Domination (-SDM) problem is to check whether has a -secure dominating set of size at most The computational complexity of -SDM has been shown to be NP-complete for split graphs and bipartite graphs [7]. The Minimum -Secure Dominating Set (M2SDS) problem is to find a -secure dominating set of minimum size in the input graph.

Preliminaries:

A vertex is a maximum neighbor of in if holds for each . A vertex is called doubly simplicial if it is a simplicial vertex and it has a maximum neighbor in . An ordering of the vertices of is a doubly perfect elimination ordering (DPEO) of if is a doubly simplicial vertex of the induced subgraph for every , . A graph is doubly chordal if and only if has a DPEO [13]. A tree is an undirected graph in which any two vertices are connected by exactly one path. A star is a tree , where and . A comb is a tree , where and . A bipartite graph is called tree convex bipartite graph if there is an associated tree such that for each vertex in , its neighborhood induces a subtree of [11]. Further if is a star or comb, then is called as star convex bipartite or comb convex bipartite respectively. A vertex cover of an undirected graph is a subset of vertices such that if edge , then either or or both.

2 Complexity Results

It is known that -SDM problem is NP-complete even for bipartite graphs and split graphs [7]. We continue investigating its NP-completeness in other special graphs. In particular, we prove that it is also NP-complete for planar graphs and doubly chordal graphs. To show that the -SDM for planar graphs is NP-complete, we use Vertex Cover problem which is NP-complete even for planar graphs [9], and is defined as follows.
Vertex Cover Decision Problem (Vertex-Cover)
Instance: A simple, undirected planar graph and a positive integer .
Question: Does there exist a vertex cover of size at most in ?

Theorem 1.

-SDM is NP-complete for planar graphs.

Proof.

Suppose a set , such that is given as a witness to a yes instance. It can be verified in polynomial time that is a -SDS of . Hence -SDM is in NP.

We reduce from Vertex-Cover problem to -SDM for planar graphs. We claim that has a vertex cover of size at most if and only if has a -SDS of size at most . ∎

To show that the -SDM is NP-complete for doubly chordal graphs, we use a well known NP-complete problem, called Exact Cover by 3-Sets (X3C) [8], which is defined as follows.
Exact Cover By 3-Sets (X3C)
Instance: A finite set with and a collection of 3-element subsets
                 of .
Question: Does contain an exact cover for , that is, a sub collection
               such that every element in occurs in exactly one member of ?

Theorem 2.

-SDM is NP-complete for doubly chordal graphs.

Proof.

It is known that the -SDM is a member of NP. To show that it is NP-complete, we propose a polynomial time reduction from X3C. We claim that the given instance of X3C has an exact cover if and only if the constructed graph has a -SDS of size at most . ∎

2.1 Complexity in some subclasses of bipartite graphs

To prove the following theorem, we use a restricted version of Exact Cover by 3-Sets, which we denote by RX3C.
Restricted Exact Cover by 3-Sets (RX3C)
Instance: A set with and a collection of 3-element subsets of
         with and each element in occurs in at most subsets.
Question: Does contain an exact cover for ?

Theorem 3.

-SDM is NP-complete for star convex bipartite graphs.

Proof.

Clearly -SDM is in NP. The proof is by reduction from RX3C problem. We claim that RX3C instance has a solution if and only if has a -SDS of size at most . ∎

Theorem 4.

-SDM is NP-complete for comb convex bipartite graphs.

Proof.

Clearly -SDM is in NP. We transform an instance of X3C problem to an instance of -SDM for comb convex bipartite graphs. Next we show that X3C instance has a solution if and only if has a -SDS of size at most . ∎

2.2 Parameterized Complexity

Now, we investigate the parameterized complexity of -SDM problem for split graphs. In [7], -SDM has been proved as NP-complete for split graphs. The decision version of domination problem is defined as follows.
Dominating Set Decision Problem (DM)
Instance: A simple, undirected graph and a positive integer .
Question: Does there exist a dominating set of size at most in ?
In [18], the DM problem has been proved as W[2]-complete, even when restricted to split graphs.

Theorem 5.

-SDM is W[2]-hard for split graphs.

Proof.

The proof is by reduction from DM problem. We show that has a dominating set of size at most if and only if has a -SDS of size at most . ∎

Since split graphs form a proper subclass of chordal graphs, the following corollary is immediate.

Corollary 1.

-SDM is W[2]-hard for chordal graphs.

2.3 Complexity in bounded tree-width graphs

Let be a graph, be a tree and be a family of vertex sets indexed by the vertices of . The pair is called a tree-decomposition of if it satisfies the following three conditions: (i) , (ii) for every edge there exists a such that both ends of lie in , (iii) whenever , , and is on the path in from to . The width of is the number , and the tree-width of is the minimum width of any tree-decomposition of . By Courcelle’s Thoerem, it is well known that every graph problem that can be described by counting monadic second-order logic (CMSOL) can be solved in linear-time in graphs of bounded tree-width, given a tree decomposition as input [6]. We show that -SDM problem can be expressed in CMSOL.

Theorem 6 (Courcelle’s Theorem).

([6]) Let be a graph property expressible in CMSOL and let be a constant. Then, for any graph of tree-width at most , it can be checked in linear-time whether has property .

Theorem 7.

Given a graph and a positive integer , -SDM can be expressed in CMSOL.

Proof.

First, we present the CMSOL formula which expresses that the graph has a dominating set of size at most

where is the binary adjacency relation which holds if and only if, are two adjacent vertices of ensures that for every vertex , either or is adjacent to a vertex in and the cardinality of is at most Now, by using the above CMSOL formula we can express -SDM in CMSOL formula as follows.

-SDM(

Therefore, -SDM can be expressed in CMSOL. ∎

Now, the following result is immediate from Theorems 6 and 7.

Theorem 8.

-SDM can be solvable in linear time for bounded tree-width graphs.

3 Approximation Results

In this section, we obtain upper and lower bounds on the approximation ratio of the M2SDS problem. We also show that the M2SDS problem is APX-complete for graphs with maximum degree .

3.1 Approximation Algorithm

Here, we propose a approximation algorithm for the M2SDS problem. In this, we will make use of two known optimization problems, MINIMUM 2-DOMINATION and MINIMUM DOMINATION. The following two theorems are the approximation results which have been obtained for these two problems.

Theorem 9.

([14]) The MINIMUM k-TUPLE DOMINATION problem in a graph with maximum degree can be approximated with an approximation ratio of

Theorem 10.

([5]) The MINIMUM DOMINATION problem in a graph with maximum degree can be approximated with an approximation ratio of

By Theorems 9 and 10, let us consider APPROX-2-DOM-SET and APP-ROX-DOM-SET are the approximation algorithms to approximate the solutions for MINIMUM 2-DOMINATION and MINIMUM DOMINATION with approximation ratios of and respectively.

Now, we propose an algorithm APPROX-2SDS to produce an approximate solution for the M2SDS problem. In APPROX-2SDS, first we compute 2-dominating set of a given graph using APPROX-2-DOM-SET. Now let . By using APPROX-DOM-SET, we compute dominating set of . Let It can be easily observed that for any two vertices there exist two vertices and such that is a dominating set of Therefore, is a -SDS of

1:A simple and undirected graph
2:A -SDS of .
3: APPROX-2-DOM-SET ()
4:Let
5: APPROX-DOM-SET ()
6:
7:return
Algorithm 1 APPROX-2SDS()
Theorem 11.

The M2SDS problem in a graph with maximum degree can be approximated with an approximation ratio of

Proof.

To prove the theorem, we show that -SDS produced by our algorithm APPROX-2SDS, , is of size at most times of , i.e.,

From the algorithm,

Since the M2SDS problem in a graph with maximum degree admits an approximation algorithm that achieves the approximation ratio of , we immediately have the following corollary of Theorem 11.

Corollary 2.

The M2SDS problem is in the class of APX when the maximum degree is fixed.

3.2 Lower bound on approximation ratio

To obtain a lower bound, we provide an approximation preserving reduction from the MINIMUM DOMINATION problem, which has the following lower bound.

Theorem 12.

[4] For a graph , the MINIMUM DOMINATION problem cannot be approximated within for any unless NP DTIME, where .

Theorem 13.

For a graph , the M2SDS problem cannot be approximated within for any unless NP DTIME

Proof.

In order to prove the theorem, we propose the following approximation preserving reduction. Let , where be an instance of the MINIMUM DOMINATION problem. From this we construct an instance of M2SDS, where , and .

Let be a minimum dominating set of a graph and be a minimum -SDS of a graph It can be observed from the reduction that by using any dominating set of a -SDS of can be formed by adding and vertices to it. Hence

Let algorithm be a polynomial time approximation algorithm to solve the M2SDS problem on graph with an approximation ratio for some fixed Let be a fixed positive integer. Next, we propose the following algorithm, DOM-SET-APPROX to find a dominating set of a given graph .

1:A simple and undirected graph
2:A dominating set of .
3:if there exists a dominating set of size at most  then
4:     
5:else
6:     Construct the graph
7:     Compute a -SDS of by using algorithm
8:           
9:     if  and  then
10:         
11:     end if     
12:     if  and  then
13:         
14:     end if
15:end if
16:return
Algorithm 2 DOM-SET-APPROX()

The algorithm DOM-SET-APPROX runs in polynomial time. It can be noted that if is a minimum dominating set of size at most , then it is optimal. Next, we analyze the case where is not a minimum dominating set of size at most

Figure 1: Example construction of a graph

Let be a minimum -SDS of , then Given a graph , DOM-SET-APPROX computes a dominating set of size . Therefore, DOM-SET-APPROX approximates a dominating set within a ratio If then the approximation ratio , where

By Theorem 12, if the MINIMUM DOMINATION problem can be approximated within a ratio of then . Similarly, if the M2SDS problem can be approximated within a ratio of then . For large values of , , for a graph where M2SDS problem cannot be approximated within a ratio of unless

3.3 APX-completeness

In this subsection, we prove that the M2SDS problem is APX-complete for graphs with maximum degree . This can be proved using an L-reduction, which is defined as follows.

definition 1.

(L-reduction) Given two NP optimization problems and and a polynomial time transformation from instances of to instances of , one can say that is an L-reduction if there exists positive constants and such that for every instance of

  1. .

  2. for every feasible solution of with objective value in polynomial time one can find a solution of with such that

Here, represents the size of an optimal solution for an instance of an NP optimization problem .

An optimization problem is APX-complete if:

  1. APX, and

  2. APX-hard, i.e., there exists an L-reduction from some known APX-complete problem to .

By Theorem 11, it is known that the M2SDS problem can be approximated within a constant factor for graphs with maximum degree . Thus, M2SDS problem is in APX for graphs with maximum degree . To show APX-hardness of M2SDS, we give an L-reduction from MINIMUM DOMINATING SET problem in graphs with maximum degree (DOM-) which has been proved as APX-complete [1].

Theorem 14.

The M2SDS problem is APX-complete for graphs with maximum degree

Proof.

It is known that M2SDS is in APX. Given an instance of DOM-, where , we construct an instance of M2SDS where and & Note that is a graph with maximum degree . An example construction of a graph from a graph is shown in Figure 2.

Figure 2: Construction of from
claim 1.

If is a minimum dominating set of and is a minimum -SDS of then where

Proof of claim.

Suppose is a minimum dominating set of , then is a -SDS of Further, if is a minimum -SDS of , then it is clear that

Next, we show that Let be any -SDS of . It is clear that for any where . Let is not a dominating set of Then there exists a vertex which is not dominated by and consequently, two attacks simultaneously happen at vertices where cannot be defended by which is a contradiction. Therefore, for every vertex there exists a vertex such that . Hence is a dominating set of and , which implies Since , it is clear that

Let and be a minimum dominating set and a minimum -SDS of and respectively. It is known that for any graph with maximum degree , , where Thus, From Claim 1 it is evident that,

Now, consider a -SDS of . Clearly, there exists a dominating set in of size at most Therefore, Hence, This proves that there is an L-reduction with and

4 Complexity difference in domination and -secure domination

Although -secure domination is one of the several variants of domination problem, however they differ in computational complexity. In particular, there exist graph classes for which the first problem is polynomial-time solvable whereas the second problem is NP-complete and vice versa. Similar study has been performed between domination and other domination parameters in [10, 16].

The DOMINATION problem is linear time solvable for doubly chordal graphs [3], but the -SDM problem is NP-complete for this class of graphs which is proved in section 2. Now, we construct a class of graphs in which the -SDM problem can be solved trivially, whereas the DOMINATION problem is NP-complete.

definition 2.

(GS graph) A graph is GS graph if it can be constructed from a connected graph where in the following way:
1. Create star graphs each with vertices, such that as the central vertex and as leaves of .
2. Attach graph and by joining to , where .

Theorem 15.

If is a GS graph obtained from a graph , then .

Proof.

Let be a GS graph. An example construction of GS graph is illustrated in Figure 3. Let It can be observed that is a -SDS of of size and hence

Let be any -set in . It is clear that for each . If for some , , then Thus, , for Hence

Figure 3: GS graph construction
lemma 1.

Let be a GS graph constructed from a graph Then has a dominating set of size at most if and only if has a dominating set of size at most

Proof.

Suppose is a dominating set of of size at most then it is clear that is a dominating set of of size at most Conversely, suppose is a dominating set of of size Clearly for each . Let be the set formed by replacing all ’s in with corresponding ’s. Clearly, is a dominating set of and . ∎

The following result is well known for the DOMINATION problem.

Theorem 16.

([8]) The DOMINATION problem is NP-complete for general graphs.

Theorem 17.

The DOMINATION problem is NP-complete for GS graphs.

Proof.

The proof directly follows from Theorem 16 and Lemma 1. ∎

It is identified that the two problems, DOMINATION and -SDM are not equivalent in computational complexity aspects. For example, when the input graph is either doubly chordal or a GS graph then complexities differ. Thus, there is a scope to study each of these problems on its own for particular graph classes.

5 Conclusion

In this paper, we have proved the NP-completeness of -SDM for planar graphs, doubly chordal graphs, star convex bipartite and comb convex bipartite graphs. On the positive side, we have proved that a minimum cardinality -secure dominating set of a graph with bounded tree-width can be computed in linear time. From approximation point of view, we have proposed an approximation algorithm for obtaining -SDS for general graphs. On the other side, we have also proved some approximation hardness results. It would be interesting to study the complexity of this problem in other graph classes such as interval graphs and block graphs.

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