Algorithmic Aspects of 2-Secure Domination in Graphs
aa r X i v : . [ c s . D M ] F e b Algorithmic Aspects of 2-Secure Domination inGraphs
J. Pavan Kumar and P.Venkata Subba ReddyDepartment of Computer Science and EngineeringNational Institute of TechnologyWarangal, Telangana 506004, [email protected], [email protected]
Abstract
Let G ( V, E ) be a simple, undirected and connected graph. A domi-nating set S ⊆ V ( G ) is called a 2- secure dominating set (2-SDS) in G ,if for every pair of distinct vertices u , u ∈ V ( G ) there exists a pairof distinct vertices v , v ∈ S such that v ∈ N [ u ], v ∈ N [ u ] and( S \ { v , v } ) ∪ { u , u } is a dominating set in G . The 2 -secure dom-ination number denoted by γ s ( G ), equals the minimum cardinality ofa 2-SDS in G . Given a graph G and a positive integer k, the 2-SecureDomination (2-SDM) problem is to check whether G has a 2-secure dom-inating set of size at most k. It is known that 2-SDM is NP-completefor bipartite graphs. In this paper, we prove that the 2-SDM problemis NP-complete for planar graphs and doubly chordal graphs, a sub-class of chordal graphs. We strengthen the NP-complete result for bipar-tite graphs, by proving this problem is NP-complete for some subclassesof bipartite graphs namely, star convex bipartite, comb convex bipar-tite graphs. We prove that 2-SDM is linear time solvable for boundedtree-width graphs. We also show that the 2-SDM is W[2]-hard even forsplit graphs. The Minimum 2-Secure Dominating Set (M2SDS) prob-lem is to find a 2-secure dominating set of minimum size in the inputgraph. We propose a ∆( G ) + 1 − approximation algorithm for M2SDS,where ∆( G ) is the maximum degree of the input graph G and prove thatM2SDS cannot be approximated within (1 − ǫ ) ln( | V | ) for any ǫ > NP ⊆ DT IME ( | V | O (log log | V | ) ). Finally, we show that the M2SDS isAPX-complete for graphs with ∆( G ) = 4 . Keywords:
Let G ( V, E ) be a simple, undirected and connected graph. For graph theoreticterminology we refer to [19]. For a vertex v ∈ V , the open neighborhood of v in1 is N G ( v )= { u ∈ V : ( u, v ) ∈ E } , the closed neighborhood of v is defined as N G [ v ] = N G ( v ) ∪ { v } . If S ⊆ V , then the open neighborhood of S is the set N G ( S ) = ∪ v ∈ S N G ( v ). The closed neighborhood of S is N G [ S ] = S ∪ N G ( S ).Let S ⊆ V . Then a vertex w ∈ V is called a private neighbor of v with respectto S if N [ w ] ∩ S = { v } . If further w ∈ V \ S , then w is called an external privateneighbor ( epn ) of v .A subset S of V is a dominating set in G if for every u ∈ V \ S , there exists v ∈ S such that ( u, v ) ∈ E . The domination number of G is the minimumcardinality of a dominating set in G and is denoted by γ ( G ). A set D is a2- dominating set if every vertex in V \ D has at least 2 neighbors in D . Adominating set S ⊆ V is said to be a secure dominating set (SDS) in G if forevery u ∈ V \ S there exists v ∈ S such that ( u, v ) ∈ E and ( S \ { v } ) ∪ { u } is adominating set of G . In this context, we say v S - defends u or u is S - defendedby v . The minimum cardinality of a SDS in G is called the secure dominationnumber of G and is denoted by γ s ( G ). Suppose a guard at a vertex v of thegraph can deal with a problem in its closed neighborhood. A dominating set S is said to be secure dominating set if an attack occurs at any vertex u ofthe graph can be S -defended by some vertex v ∈ S in its closed neighborhood.However, suppose if two attacks simultaneously happen at any two vertices ofthe graph, then how to defend both the vertices is an interesting problem. Adominating set S ⊆ V ( G ) is called a 2- secure dominating set (2-SDS) in G , iffor every pair of distinct vertices u , u ∈ V ( G ) there exists a pair of distinctvertices v , v ∈ S such that v ∈ N [ u ], v ∈ N [ u ] and ( S \ { v , v } ) ∪ { u , u } is a dominating set in G . The 2 -secure domination number denoted by γ s ( G ),equals the minimum cardinality of a 2-SDS in G and any minimum 2-securedominating set is referred as γ s -set of G . Given a graph G and a positiveinteger k, the 2-Secure Domination (2-SDM) problem is to check whether G has a 2-secure dominating set of size at most k. The computational complexityof 2-SDM has been shown to be NP-complete for split graphs and bipartitegraphs [7]. The Minimum 2-Secure Dominating Set (M2SDS) problem is to finda 2-secure dominating set of minimum size in the input graph.
Preliminaries:
A vertex u ∈ N [ v ] is a maximum neighbor of v in G if N [ w ] ⊆ N [ u ] holds for each w ∈ N [ v ]. A vertex v ∈ V ( G ) is called doubly simplicial if it is a simplicial vertex and it has a maximum neighbor in G . An ordering { v , v , . . . , v n } of the vertices of V ( G ) is a doubly perfect elimination ordering (DPEO) of G if v i is a doubly simplicial vertex of the induced subgraph G i = G [ { v i , v i +1 , . . . , v n } ] for every i , 1 ≤ i ≤ n . A graph G is doubly chordal ifand only if G has a DPEO [13]. A tree is an undirected graph in which anytwo vertices are connected by exactly one path. A star is a tree T = ( A, F ),where A = { a , a , . . . , a n } and F = { ( a , a i ) : 1 ≤ i ≤ n } . A comb is a tree T = ( A, F ), where A = { a , a , . . . , a n } and F = { ( a i , a i +1 ) : 1 ≤ i ≤ n − }∪ { ( a i , a n + i ) : 1 ≤ i ≤ n } . A bipartite graph G ( A, B, E ) is called tree convexbipartite graph if there is an associated tree T = ( A, F ) such that for eachvertex b in B , its neighborhood N G ( b ) induces a subtree of T [11]. Further if2 is a star or comb, then G is called as star convex bipartite or comb convexbipartite respectively. A vertex cover of an undirected graph G ( V, E ) is a subsetof vertices V ′ ⊆ V such that if edge ( u, v ) ∈ E , then either u ∈ V ′ or v ∈ V ′ orboth. It is known that 2-SDM problem is NP-complete even for bipartite graphs andsplit graphs [7]. We continue investigating its NP-completeness in other specialgraphs. In particular, we prove that it is also NP-complete for planar graphsand doubly chordal graphs. To show that the 2-SDM for planar graphs is NP-complete, we use Vertex Cover problem which is NP-complete even for planargraphs [9], and is defined as follows.
Vertex Cover Decision Problem (Vertex-Cover)
Instance:
A simple, undirected planar graph G and a positive integer k . Question:
Does there exist a vertex cover of size at most k in G ? Theorem 1. -SDM is NP-complete for planar graphs.Proof. Suppose a set S ⊆ V , such that | S | ≤ k is given as a witness to a yesinstance. It can be verified in polynomial time that S is a 2-SDS of G . Hence2-SDM is in NP.We reduce from Vertex-Cover problem to 2-SDM for planar graphs. Weclaim that G has a vertex cover of size at most k if and only if G ∗ has a 2-SDSof size at most r = m + n + k + 2.To show that the 2-SDM is NP-complete for doubly chordal graphs, we usea 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 X with | X | = 3 q and a collection C of 3-element subsetsof X . Question:
Does C contain an exact cover for X , that is, a sub collection C ′ ⊆ C such that every element in X occurs in exactly one member of C ′ ? Theorem 2. -SDM is NP-complete for doubly chordal graphs.Proof. It is known that the 2-SDM is a member of NP. To show that it is NP-complete, we propose a polynomial time reduction from X3C. We claim thatthe given instance of X3C < X, C > has an exact cover if and only if theconstructed graph G has a 2-SDS of size at most l = q + 2. To prove the following theorem, we use a restricted version of Exact Cover by3-Sets, which we denote by RX3C. 3 estricted Exact Cover by 3-Sets (RX3C)
Instance:
A set X with | X | = 3 q and a collection C of 3-element subsets of X with | C | = m > q and each element in X occurs in at most q subsets. Question:
Does C contain an exact cover for X ? Theorem 3. -SDM is NP-complete for star convex bipartite graphs.Proof. Clearly 2-SDM is in NP. The proof is by reduction from RX3C problem.We claim that RX3C instance < X, C > has a solution C ′ if and only if G hasa 2-SDS of size at most q + 8. Theorem 4. -SDM is NP-complete for comb convex bipartite graphs.Proof. Clearly 2-SDM is in NP. We transform an instance of X3C problem toan instance of 2-SDM for comb convex bipartite graphs. Next we show thatX3C instance < X, C > has a solution C ′ if and only if G has a 2-SDS of sizeat most q + 8. Now, we investigate the parameterized complexity of 2-SDM problem for splitgraphs. In [7], 2-SDM has been proved as NP-complete for split graphs. Thedecision version of domination problem is defined as follows.
Dominating Set Decision Problem (DM)
Instance:
A simple, undirected graph G ( V, E ) and a positive integer k . Question:
Does there exist a dominating set of size at most k in G ?In [18], the DM problem has been proved as W[2]-complete, even when restrictedto split graphs. Theorem 5. -SDM is W[2]-hard for split graphs.Proof. The proof is by reduction from DM problem. We show that G has adominating set of size at most k if and only if G ∗ has a 2-SDS of size at most r = k + 2.Since split graphs form a proper subclass of chordal graphs, the following corol-lary is immediate. Corollary 1. -SDM is W[2]-hard for chordal graphs. Let G be a graph, T be a tree and v be a family of vertex sets V t ⊆ V ( G )indexed by the vertices t of T . The pair ( T, v ) is called a tree-decompositionof G if it satisfies the following three conditions: (i) V ( G ) = S t ∈ V ( T ) V t , (ii)for every edge e ∈ E ( G ) there exists a t ∈ V ( T ) such that both ends of e lie in V t , (iii) V t ∩ V t ⊆ V t whenever t , t , t ∈ V ( T ) and t is on the path in T t to t . The width of ( T, v ) is the number max {| V t | − t ∈ T } , andthe tree-width tw ( G ) of G is the minimum width of any tree-decomposition of G . By Courcelle’s Thoerem, it is well known that every graph problem that canbe described by counting monadic second-order logic (CMSOL) can be solvedin linear-time in graphs of bounded tree-width, given a tree decomposition asinput [6]. We show that 2-SDM problem can be expressed in CMSOL. Theorem 6 ( Courcelle’s Theorem ) . ( [6]) Let P be a graph property expressiblein CMSOL and let k be a constant. Then, for any graph G of tree-width at most k , it can be checked in linear-time whether G has property P . Theorem 7.
Given a graph G and a positive integer k , -SDM can be expressedin CMSOL.Proof. First, we present the CMSOL formula which expresses that the graph G has a dominating set of size at most k. Dominating ( S ) = ( | S | ≤ k ) ∧ ( ∀ p )(( ∃ q )( q ∈ S ∧ adj ( p, q ))) ∨ ( p ∈ S ) where adj ( p, q ) is the binary adjacency relation which holds if and only if, p, q are two adjacent vertices of G. Dominating ( S ) ensures that for every vertex p ∈ V , either p ∈ S or p is adjacent to a vertex in S and the cardinality of S isat most k. Now, by using the above CMSOL formula we can express 2-SDM inCMSOL formula as follows. S ) = Dominating ( S ) ∧ ( ∀ x )( ∀ y )(( ∃ p )( ∃ p )( p ∈ N [ x ] ∧ p ∈ N [ y ] \ { p }∧ Dominating (( S \ { x, y } ) ∪ { p , p } )) Therefore, 2-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-widthgraphs. In this section, we obtain upper and lower bounds on the approximation ratio ofthe M2SDS problem. We also show that the M2SDS problem is APX-completefor graphs with maximum degree 4.
Here, we propose a ∆( G ) + 1 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 theoremsare the approximation results which have been obtained for these two problems.5 heorem 9. ( [14]) The MINIMUM k-TUPLE DOMINATION problem in agraph with maximum degree ∆( G ) can be approximated with an approximationratio of G ) + 1) . Theorem 10. ( [5]) The MINIMUM DOMINATION problem in a graph withmaximum degree ∆( G ) can be approximated with an approximation ratio of G ) + 1) . By Theorems 9 and 10, let us consider APPROX-2-DOM-SET and APP-ROX-DOM-SET are the approximation algorithms to approximate the solutionsfor MINIMUM 2-DOMINATION and MINIMUM DOMINATION with approx-imation ratios of 1 + ln(∆( G ) + 1) and 1 + ln(∆( G ) + 1) respectively.Now, we propose an algorithm APPROX-2SDS to produce an approximatesolution for the M2SDS problem. In APPROX-2SDS, first we compute 2-dominating set D of a given graph G using APPROX-2-DOM-SET. Now let G ′ = G [ V \ D ]. By using APPROX-DOM-SET, we compute dominating set D ′ of G ′ . Let D = D ∪ D ′ . It can be easily observed that for any two vertices u , u ∈ V there exist two vertices v ∈ D ∩ N [ u ] and v ∈ D ∩ N [ u ] such that( D \ { v , v } ) ∪ { u , u } is a dominating set of G. Therefore, D is a 2-SDS of G. Algorithm 1
APPROX-2SDS( G ) Input:
A simple and undirected graph G Output:
A 2-SDS D of G . D ← APPROX-2-DOM-SET ( G ) Let G ′ = G [ V \ D ] D ′ ← APPROX-DOM-SET ( G ′ ) D ← D ∪ D ′ return D. Theorem 11.
The M2SDS problem in a graph G with maximum degree ∆( G ) can be approximated with an approximation ratio of ∆( G ) + 1 . Proof.
To prove the theorem, we show that 2-SDS produced by our algorithmAPPROX-2SDS, D , is of size at most (∆( G ) + 1) times of γ s ( G ), i.e., | D | ≤ (∆( G ) + 1) γ s ( G )From the algorithm, | D | = | D ∪ D ′ | = | D | + | D ′ | ≤ n ≤ (∆( G ) + 1) γ ( G ) ≤ (∆( G ) + 1) γ s ( G )6ince the M2SDS problem in a graph with maximum degree ∆( G ) admitsan approximation algorithm that achieves the approximation ratio of ∆( G ) + 1,we immediately have the following corollary of Theorem 11. Corollary 2.
The M2SDS problem is in the class of APX when the maximumdegree ∆( G ) is fixed. To obtain a lower bound, we provide an approximation preserving reductionfrom the MINIMUM DOMINATION problem, which has the following lowerbound.
Theorem 12. [4] For a graph G ( V, E ) , the MINIMUM DOMINATION prob-lem cannot be approximated within (1 − ǫ ) ln n for any ǫ > unless NP ⊆ DTIME ( n O (log log n ) ) , where n = | V | . Theorem 13.
For a graph G ( V, E ) , the M2SDS problem cannot be approxi-mated within (1 − ǫ ) ln | V | for any ǫ > unless NP ⊆ DTIME ( | V | O (log log | V | ) ) . Proof.
In order to prove the theorem, we propose the following approximationpreserving reduction. Let G ( V, E ), where V = { v , v , . . . , v n } be an instance ofthe MINIMUM DOMINATION problem. From this we construct an instance G ′ ( V ′ , E ′ ) of M2SDS, where V ′ = V ∪{ w , w , z , z , z } , and E ′ = E ∪{ ( v i , w ) , ( v i , w ) : v i ∈ V } ∪ { ( w , z )( w , z ) , ( z , z ) } .Let D ∗ be a minimum dominating set of a graph G and S ∗ be a minimum2-SDS of a graph G ′ . It can be observed from the reduction that by using anydominating set of G, a 2-SDS of G ′ can be formed by adding w , w and z vertices to it. Hence | S ∗ | ≤ | D ∗ | + 3 . Let algorithm A be a polynomial time approximation algorithm to solve theM2SDS problem on graph G ′ with an approximation ratio α = (1 − ǫ ) ln | V ′ | for some fixed ǫ > . Let k be a fixed positive integer. Next, we propose thefollowing algorithm, DOM-SET-APPROX to find a dominating set of a givengraph G .The algorithm DOM-SET-APPROX runs in polynomial time. It can benoted that if D is a minimum dominating set of size at most k , then it isoptimal. Next, we analyze the case where D is not a minimum dominating setof size at most k. lgorithm 2 DOM-SET-APPROX( G ) Input:
A simple and undirected graph G Output:
A dominating set D of G . if there exists a dominating set D ′ of size at most k then D ← D ′ else Construct the graph G ′ Compute a 2-SDS S of G ′ by using algorithm A D ← S ∩ V if w ∈ S and ∃ v u , v u ∈ V \ N [ D ] then D ← D ∪ { v u } end if if w , z ∈ S and ∃ v t , v t ∈ V \ N [ D ] then D ← D ∪ { v t } end if end if return D. • v • v • v • v • v • v • w • w • z • z • z Figure 1: Example construction of a graph G ′ Let S ∗ be a minimum 2-SDS of G ′ , then | S ∗ | ≥ k. Given a graph G , DOM-SET-APPROX computes a dominating set of size | D | ≤ | S | ≤ α | S ∗ | ≤ α ( | D ∗ | +3) = α (1 + 3 / | D ∗ | ) | D ∗ | ≤ α (1 + 3 /k ) | D ∗ | . Therefore, DOM-SET-APPROXapproximates a dominating set within a ratio α (1 + 3 /k ) . If 3 /k < ǫ/ , thenthe approximation ratio α (1 + 3 /k ) < (1 − ǫ )(1 + ǫ/
2) ln n = (1 − ǫ ′ ) ln n , where ǫ ′ = ǫ/ ǫ / . By Theorem 12, if the MINIMUM DOMINATION problem can be approxi-mated within a ratio of (1 − ǫ ′ ) ln n, then N P ⊆ DT IM E ( n O (log log n ) ). Simi-larly, if the M2SDS problem can be approximated within a ratio of (1 − ǫ ) ln n, then N P ⊆ DT IM E ( n O (log log n ) ). For large values of n , ln n ≅ ln( n + 5), for a8raph G ′ ( V ′ , E ′ ) , where | V ′ | = | V | +5 , M2SDS problem cannot be approximatedwithin a ratio of (1 − ǫ ) ln | V ′ | unless N P ⊆ DT IM E ( | V ′ | O (log log | V ′ | ) ) . In this subsection, we prove that the M2SDS problem is APX-complete forgraphs with maximum degree 4. This can be proved using an L-reduction,which is defined as follows. definition 1. (L-reduction)
Given two NP optimization problems F and G and a polynomial time transformation f from instances of F to instances of G ,one can say that f is an L-reduction if there exists positive constants α and β such that for every instance x of F opt G ( f ( x )) ≤ α.opt F ( x ) .2. for every feasible solution y of f ( x ) with objective value m G ( f ( x ) , y ) = c in polynomial time one can find a solution y ′ of x with m F ( x, y ′ ) = c such that | opt F ( x ) − c | ≤ β | opt G ( f ( x )) − c | . Here, opt F ( x ) represents the size of an optimal solution for an instance x of anNP optimization problem F . An optimization problem π is APX-complete if:1. π ∈ APX, and2. π ∈ APX-hard, i.e., there exists an L-reduction from some known APX-complete problem to π .By Theorem 11, it is known that the M2SDS problem can be approximatedwithin a constant factor for graphs with maximum degree 4. Thus, M2SDSproblem is in APX for graphs with maximum degree 4. To show APX-hardnessof M2SDS, we give an L-reduction from MINIMUM DOMINATING SET prob-lem in graphs with maximum degree 3 (DOM-3) which has been proved asAPX-complete [1]. Theorem 14.
The M2SDS problem is APX-complete for graphs with maximumdegree . Proof.
It is known that M2SDS is in APX. Given an instance G ( V, E ) of DOM-3, where V = { v , v , . . . , v n } , we construct an instance G ′ ( V ′ , E ′ ) of M2SDSwhere V ′ = V ∪ { x i , x i , x i : 1 ≤ i ≤ ⌈ n ⌉} and E ′ = E ∪ { ( v i , x i +1) / ) , ( v i +1 , x i +1) / ) : 1 ≤ i ≤ n − i = 1 (mod 2) } ∪ { ( v n , x n +1) / ) : n = 1(mod 2) } ∪ { ( x i , x i ) , ( x i , x i ) : 1 ≤ i ≤ ⌈ n ⌉} . Note that G ′ is a graph withmaximum degree 4. An example construction of a graph G ′ from a graph G isshown in Figure 2. claim 1. If D ∗ is a minimum dominating set of G and S ∗ is a minimum -SDSof G ′ then | S ∗ | = | D ∗ | + 2 ⌈ n ⌉ , where n = | V | . • v • v • x • x • x • v • x • x • v • x Figure 2: Construction of G ′ from G Proof of claim.
Suppose D ∗ is a minimum dominating set of G , then D ∗ ∪{ x i , x i : 1 ≤ i ≤ ⌈ n ⌉} is a 2-SDS of G ′ . Further, if S ∗ is a minimum 2-SDS of G ′ , then it is clear that | S ∗ | ≤ | D ∗ | + 2 ⌈ n ⌉ . Next, we show that | S ∗ | ≥ | D ∗ | + 2 ⌈ n ⌉ . Let S be any 2-SDS of G ′ . It is clearthat for any i, where 1 ≤ i ≤ ⌈ n ⌉ , | S ∩ { x i , x i , x i }| ≥
2. Let D = S ∩ V is nota dominating set of G. Then there exists a vertex v j which is not dominated by D and consequently, two attacks simultaneously happen at vertices v j , v k where v k ∈ N ( v j ) \ V cannot be defended by S, which is a contradiction. Therefore, forevery vertex u ∈ V \ D there exists a vertex v ∈ D such that ( u, v ) ∈ E . Hence D is a dominating set of G and | D | ≥ | D ∗ | , which implies | S | ≥ | D ∗ | + 2 ⌈ n ⌉ . Since | S | ≥ | S ∗ | , it is clear that | S ∗ | ≥ | D ∗ | + 2 ⌈ n ⌉ . Let D ∗ and S ∗ be a minimum dominating set and a minimum 2-SDS of G and G ′ respectively. It is known that for any graph H with maximum degree∆( H ), γ ( H ) ≥ n ∆( H )+1 , where n = | V ( H ) | . Thus, | D ∗ | ≥ n . From Claim 1 it isevident that, | S ∗ | = | D ∗ | + 2 ⌈ n ⌉ ≤ | D ∗ | + n + 1 ≤ | D ∗ | + 4 | D ∗ | + 1 ≤ | D ∗ | . Now, consider a 2-SDS S of G ′ . Clearly, there exists a dominating set D in G of size at most | S | − ⌈ n ⌉ . Therefore, | D | ≤ | S | − ⌈ n ⌉ . Hence, | D | − | D ∗ | ≤| S | − ⌈ n ⌉ − | D ∗ | = | S | − | S ∗ | . This proves that there is an L-reduction with α = 6 and β = 1 . -secure domination Although 2-secure domination is one of the several variants of domination prob-lem, however they differ in computational complexity. In particular, there existgraph classes for which the first problem is polynomial-time solvable whereasthe second problem is NP-complete and vice versa. Similar study has beenperformed between domination and other domination parameters in [10, 16].The DOMINATION problem is linear time solvable for doubly chordal graphs[3], but the 2-SDM problem is NP-complete for this class of graphs which is10roved in section 2. Now, we construct a class of graphs in which the 2-SDMproblem 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 froma connected graph G ( V, E ) where | V | = n, in the following way:1. Create n star graphs { S , S , . . . , S n } each with vertices, such that b i as thecentral vertex and a i , c i , d i as leaves of S i .2. Attach graph G and S i by joining v i to a i , where ≤ i ≤ n . Theorem 15. If G ′ is a GS graph obtained from a graph G ( V, E ) ( | V | = n ) ,then γ s ( G ′ ) = 3 n .Proof. Let G ′ ( V ′ , E ′ ) be a GS graph. An example construction of GS graph isillustrated in Figure 3. Let S = { v i , b i , c i : 1 ≤ i ≤ n } . It can be observed that S is a 2-SDS of G ′ of size 3 n and hence γ s ( G ′ ) ≤ n. Let S ∗ be any γ s -set in G ′ . It is clear that | S ∗ ∩ { b i , c i , d i }| ≥ , for each1 ≤ i ≤ n . If for some i , S ∗ ∩ { v i , a i } = ∅ , then | S ∗ ∩ { b i , c i , d i }| = 3 . Thus, | S ∗ ∩ { v i , a i , b i , c i , d i }| ≥
3, for 1 ≤ i ≤ n. Hence γ s ( G ′ ) ≥ n. • v • a • b • c • d • v • a • b • c • d • a • b • c • d • v • a • b • d • c • v G Figure 3: GS graph construction lemma 1.
Let G ′ be a GS graph constructed from a graph G ( V, E ) . Then G hasa dominating set of size at most k if and only if G ′ has a dominating set of sizeat most k + n. Proof.
Suppose D is a dominating set of G of size at most k, then it is clear that D ∪ { b i : 1 ≤ i ≤ n } is a dominating set of G ′ of size at most k + n. Conversely,suppose D ′ is a dominating set of G ′ of size k + n. Clearly | D ′ ∩{ a i , b i , c i , d i }| ≥ , for each 1 ≤ i ≤ n . Let D ′′ be the set formed by replacing all a i ’s in D ′ withcorresponding v i ’s. Clearly, D ′′ is a dominating set of G and | D ′′ | ≤ k .The following result is well known for the DOMINATION problem. Theorem 16. ( [8]) The DOMINATION problem is NP-complete for generalgraphs. heorem 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 2-SDM are not equiv-alent in computational complexity aspects. For example, when the input graphis either doubly chordal or a GS graph then complexities differ. Thus, there isa scope to study each of these problems on its own for particular graph classes.
In this paper, we have proved the NP-completeness of 2-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 2-secure dom-inating set of a graph with bounded tree-width can be computed in linear time.From approximation point of view, we have proposed an approximation algo-rithm for obtaining 2-SDS for general graphs. On the other side, we have alsoproved some approximation hardness results. It would be interesting to studythe complexity of this problem in other graph classes such as interval graphsand block graphs.
References [1] P. Alimonti, and V. Kann,
Some APX-completeness results for cubicgraphs , Theoret. Comp. Sci. (2000) 123-134.doi:10.1016/S0304-3975(98)00158-3[2] A.A. Bertossi,
Dominating sets for split and bipartite graphs , Inform. Proc.Let., (1984) 37–40.[3] A. Brandstdt, V.D. Chepoi, and F.F. Dragan,
The algorithmic use of hy-pertree structure and maximum neighbourhood orderings , Disc. Appl. Math. (1998) 43-77. doi: 10.1016/S0166-218X(97)00125-X[4] M. Chleb´ık and J. Chleb´ıko´va, Approximation hardness of dominating setproblems in bounded degree graphs , Inform. and Comp. (2008) 1264-1275. doi:10.1016/j.ic.2008.07.003[5] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein, Introduction toAlgorithms (Prentice Hall, India, 2001).[6] B. Courcelle,
The monadic second-order logic of graphs. I. Recognizable setsof finite graphs , Inform. and Comp. (1990) 64-75.doi:10.1016/0890-5401(90)90043-H.[7] L. Devendra, P. Venkata Subba Reddy, and J. Pavan Kumar,
Complexityissues of variants of secure domination in graphs
Elect. Not. in Disc. Math. (2017) 77-84. 128] M.R. Garey, and D.S. Johnson, Computers and Intractability: A Guide tothe Theory of NP-Completeness (Freeman, New York, 1979).[9] M.R. Garey, and D.S. Johnson,
The rectilinear Steiner tree problem is NP-complete , SIAM J. on Appl. Math. (1977) 826-834.[10] M.A. Henning, and Arti Pandey,
Algorithmic aspects of semito-tal domination in graphs , Theoret. Comput. Sci. (2019) 46-57.doi:10.1016/j.tcs.2018.09.019[11] W. Jiang, T. Liu, T. Ren, and K. Xu,
Two Hardness Results on FeedbackVertex Sets,
In FAW-AAIM (2011) 233–243.[12] R.M. Karp,
Reducibility among combinatorial problems,
Compl. of Comp.Computations. Springer US, (1972) 85–103.[13] M. Moscarini,
Doubly chordal graphs, steiner trees, and connected domina-tion , Networks (1993) 59-69. doi:10.1002/net.3230230108[14] R. Klasing, and C. Laforest,
Hardness results and approximation algorithmsof k-tuple domination in graphs , Inform. Process. Lett. (2004) 7583.doi:10.1016/j.ipl.2003.10.004[15] B.S. Panda, and Arti Pandey, Algorithm and hardness results for outer-connected dominating set in graphs , J. of Gra. Algo. and Appl. (2014)493-513. doi:10.7155/jgaa.00334[16] J. Pavan Kumar, and P. Venkata Subba Reddy,
Algorithmic aspects ofsecure connected domination in graphs , Discuss. Math. Graph Theory.
Inpress [17] N. Mahadev, U. Peled,
Threshold Graphs and Related Topics, in: Annalsof Discrete Mathematics , vol. 56, North Holland, (1995).[18] V. Raman, and S. Saurabh.
Short cycles make W-hard problems hard: FPTalgorithms for W-hard problems in graphs with no short cycles , Algorith-mica52.2