Algorithmic Aspects of Secure Connected Domination in Graphs
aa r X i v : . [ c s . D M ] J a n Algorithmic Aspects of Secure ConnectedDomination in Graphs
Jakkepalli Pavan Kumar, P. Venkata Subba ReddyDepartment of Computer Science and EngineeringNational Institute of TechnologyWarangal, Telangana, India
Abstract
Let G = ( V , E ) be a simple, undirected and connected graph. A connected dom-inating set S ⊆ V is a secure connected dominating set of G , if for each u ∈ V \ S ,there exists v ∈ S such that ( u , v ) ∈ E and the set ( S \ { v } ) ∪ { u } is a connecteddominating set of G . The minimum size of a secure connected dominating set of G denoted by γ sc ( G ), is called the secure connected domination number of G . Givena graph G and a positive integer k , the Secure Connected Domination (SCDM)problem is to check whether G has a secure connected dominating set of size atmost k . In this paper, we prove that the SCDM problem is NP-complete for doublychordal graphs, a subclass of chordal graphs. We investigate the complexity ofthis problem for some subclasses of bipartite graphs namely, star convex bipartite,comb convex bipartite, chordal bipartite and chain graphs. The Minimum SecureConnected Dominating Set (MSCDS) problem is to find a secure connected dom-inating set of minimum size in the input graph. We propose a ( ∆ ( G ) +
1) - approx-imation algorithm for MSCDS, where ∆ ( G ) is the maximum degree of the inputgraph G and prove that MSCDS cannot be approximated within (1 ǫ ) ln( | V | ) for any ǫ > NP ⊆ DT IME ( | V | O (loglog | V | ) ) even for bipartite graphs. Finally, weshow that the MSCDS is APX-complete for graphs with ∆ ( G ) = . Throughout this paper, all graphs G = ( V , E ) should be finite, simple (i.e., without self-loops and multiple edges), undirected and connected with vertex set V and edge set E .The ( open ) neighborhood of a vertex v ∈ V is the set N ( v ) = { u ∈ V | ( u , v ) ∈ E } .If X ⊆ V , then the open neighborhood of X is the set N ( X ) = ∪ v ∈ X N ( v ). The closedneighborhood of X is N [ X ] = X ∪ N ( X ). The degree of a vertex v is | N ( v ) | and is denotedby d ( v ). If d ( v ) =
1, then v is called a pendant vertex of G and the support vertex of apendant vertex v is the unique vertex s such that ( v , s ) ∈ E . The maximum degree ofvertices in V is denoted by ∆ ( G ). For a graph G , and a set S ⊆ V , the subgraph of Ginduced by S is defined as G [ S ] = ( S , E S ), where E S = { ( x , y ) ∈ E : x , y ∈ S } . If G [ S ],where S ⊆ V , is a complete subgraph of G , then it is called a clique of G . A set S ⊆ V is an independent set if G [ S ] has no edge. A split graph is a graph in which the vertices1an be partitioned into a clique and an independent set. For undefined terminology andnotations we refer to [25].A set S ⊆ V is a dominating set (DS) in G if for every u ∈ V \ S , there exists v ∈ S such that ( u , v ) ∈ E , i.e., N [ S ] = V . The minimum size of dominating set in G is called the domination number of G and is denoted by γ ( G ). A set S ⊆ V is a connected dominating set (CDS) of G if G [ S ] is connected and every vertex not in S is adjacent to a vertex in S . The minimum size of CDS in G is called the connecteddomination number of G and is denoted by γ c ( G ). The study of domination and relatedproblems is one of the fastest growing areas in graph theory. The study of literature onvarious domination parameters in graphs has been surveyed and outlined in [12, 13].An important domination parameter called secure domination has been introduced byE.J. Cockayne et al. in [6]. A set S ⊆ V is a secure dominating set (SDS) of G if, foreach vertex u ∈ V \ S , there exists a neighboring vertex v of u in S such that ( S \{ v } ) ∪{ u } is a dominating set of G (in which case v is said to defend u ). The decision version ofsecure domination problem is known to be NP-complete for general graphs [9] andremains NP-complete even for various restricted families of graphs such as bipartite,doubly chordal and split graphs [17, 24]. Recently, H. Wang et al. [24] obtained someapproximation results related to secure domination. A CDS S of G is called a secureconnected dominating set (SCDS) in G if, for each u ∈ V \ S , there exists v ∈ S suchthat ( u , v ) ∈ E and ( S \ { v } ) ∪ { u } is a CDS in G (in which case v is said to be S - defender ). The secure connected domination number of graph G is the minimum sizeof a SCDS, and is denoted by γ sc ( G ) [4]. Given a graph G and a positive integer k , theSecure Connected Domination (SCDM) problem is to check whether G has a SCDSof size at most k . It is known that SCDM is NP-complete for bipartite graphs and splitgraphs, whereas it is linear time solvable for block graphs and threshold graphs [22].The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a SCDSof minimum size in the input graph.
Preliminaries
In a graph G , a vertex v is simplicial if its closed neighborhood N G [ v ]induces a complete subgraph of G . An ordering { v , v , . . . , v n } of the vertices of V is a perfect elimination ordering (PEO), if v i is a simplicial of the induced subgraph G i = G [ { v i , v i + , . . . , v n } ] for every i , 1 ≤ i ≤ n . A graph G is chordal if and only if G admits a PEO. 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 is called doubly simplicial if it is a simplicialvertex and has a maximum neighbor. An ordering { v , v , . . . , v n } of the vertices of V isa doubly perfect elimination ordering (DPEO) of G if v i is a doubly simplicial vertexof the induced subgraph G i = G [ { v i , v i + , . . . , v n } ] for every i , 1 ≤ i ≤ n . A graph G is doubly chordal if and only if it has a DPEO [18]. Alternatively, doubly chordalgraphs are chordal and dually chordal graphs. An undirected graph is a tree if it isconnected and cycle-free. A star is a tree T = ( A , F ), where A = { a , a , . . . , a n } and F = { ( a , a i ) | ≤ i ≤ n } . A comb is a tree T = ( A , F ), where A = { a , a , . . . , a n } and F = { ( a i , a i + ) | ≤ i ≤ n − } ∪ { ( a i , a n + i ) | ≤ i ≤ n } . A bipartite graph G = ( A , B , E )is called tree convex bipartite graph if there is an associated tree T = ( A , F ) such thatfor each vertex b in B , its neighborhood N G ( b ) induces a subtree of T [15]. If T is a star(or comb), then G is called as star convex bipartite (or comb convex bipartite ) graph. A2raph G is chordal bipartite if G is bipartite and each cycle of G of length greater than4 has a chord. Alternatively, chordal bipartite graphs are weakly chordal and bipartitegraphs. In this section, we show that the complexity of SCDM in doubly chordal, star convexbipartite, comb convex bipartite, and chordal bipartite graphs is NP-complete. Also, weprove that SCDM is linear time solvable in chain graphs, a subclass of bipartite graphs.The decision version of domination and secure connected domination problems aredefined as follows.
Domination Decision Problem (DOMINATION)
Instance:
A simple, undirected graph G and a positive integer k . Question:
Does there exist a dominating set of size at most k in G ? Secure Connected Domination Problem (SCDM)
Instance : A simple, undirected and connected graph G and a positive integer l . Question : Does there exist a SCDS of size at most l in G ?Domination decision problem for bipartite graphs has been proved as NP-complete [2].Let P ( G ) and S ( G ) be the set of pendant and support vertices of G , respectively. Proposition 1. ([4]) Let G be a connected graph of order n ≥ and let X be a secureconnected dominating set of G. Then(i) P ( G ) ⊆ X and S ( G ) ⊆ X,(ii) no vertex in P ( G ) ∪ S ( G ) is an X-defender. To prove the NP-completeness of the SCDM for doubly chordal graphs we considerthe following SET-COVER decision problem which has been proved as NP-complete[16].
Set Cover Decision Problem (SET-COVER)
Instance:
A finite set X of elements, a family of m subsets of elements C and apositive integer k . Question:
Does there exist a subfamily of k subsets C ′ whose union equals X ? Theorem 1.
SCDM is NP-complete for doubly chordal graphs.Proof.
Clearly, SCDM is in NP. If a set S ⊆ V , such that | S | ≤ l is given as a witnessto a yes instance then it can be verified in polynomial time that S is a SCDS of G .Let X = { x , x , . . . , x n } , C = { C , C , . . . , C m } be an instance of SET-COVERproblem. We now construct an instance of SCDM from the given instance of SET-COVER as similar to the reduction in [24] as follows. Construct a graph G with thefollowing vertices: (i) a vertex x i for each element x i ∈ X , (ii) vertex c j for each subset C j ∈ C and let C ∗ = { c j : 1 ≤ j ≤ m } and (iii) two vertices p and q . Add the following3dges in G : (i) if x i ∈ C j , then add edge ( x i , c j ), where 1 ≤ i ≤ n and 1 ≤ j ≤ m ,(ii) edges between every pair of vertices in the set C ∗ ∪ { p } , (iii) edges between x i and p , where 1 ≤ i ≤ n and (iv) edge between p and q . Since G admits a DPEO { x , x , . . . , x n , c , c , . . . , c m , p , q } , it is a doubly chordal graph and the construction of G can be accomplished in polynomial time.Now we show that the given instance of SET-COVER problem < X , C > has a setcover of size at most k if and only if the constructed graph G has a SCDS of size atmost l = k +
2. Suppose C ′ ⊆ C is a set cover of X , with | C ′ | ≤ k , then it is easy toverify that the set S = { c j : C j ∈ C ′ } ∪ { p , q } is a SCDS of size at most k + G .Conversely, suppose S ⊆ V is a SCDS of size at most l = k + G . FromProposition 1, it is clear that | S ∩ { p , q }| =
2. Let X ∗ = S ∩ X and S ∗ = S ∩ { c j : 1 ≤ j ≤ m } . If | X ∗ | =
0, then we are done, that is respective subsets of vertices of S ∗ form thesolution for SET-COVER and clearly | S ∗ | ≤ k . Otherwise, since X is an independentset, every vertex in X ∗ can be replaced with its adjacent vertex in the set C ∗ and size ofthe resultant set is at most k . Therefore, there exists a set cover of size at most k . (cid:3) Theorem 2.
SCDM is NP-complete for star convex bipartite graphs.Proof.
It is known that SCDM is in NP. We reduce the Domination problem for bi-partite graphs to SCDM for star convex bipartite graphs as follows. The reduction issimilar to the construction given in [24]. Given an instance G = ( A , B , E ) of Domina-tion problem for bipartite graphs, where A = { a , a , . . . , a p } and B = { b , b , . . . , b q } ,we construct an instance G ′ = ( A ′ , B ′ , E ′ ) of SCDM, where A ′ = A ∪ { a x , a y } , B ′ = B ∪ { b x , b y } , and E ′ = E ∪ { ( a x , b i ) : 1 ≤ i ≤ q } ∪ { ( b x , a i ) : 1 ≤ i ≤ p } ∪{ ( a x , b x ) , ( a x , b y ) , ( b x , a y ) } . It can be verified that G ′ is a star convex bipartite graphwith its associated star T = ( A ′ , F ), where F = { ( a x , a i ) | ≤ i ≤ p } ∪ { ( a x , a y ) } . Notethat the construction of graph G ′ can be done in polynomial time.Next we show that G has a dominating set of size at most k if and only if G ′ has aSCDS of size at most l = k +
4. Suppose D is a dominating set in G of size at most k .Then it can be easily verified that the set D ∪ { a x , a y , b x , b y } is a SCDS in G ′ of size atmost k + S be a SCDS in G ′ with | S | ≤ l = k +
4. From Proposition 1,it is clear that | S ∩{ a x , a y , b x , b y }| =
4. Let S ∗ = S \ { a x , a y , b x , b y } . Thus, | S ∗ | ≤ k . Since S is a SCDS and a x and b x are support vertices, for every vertex a i ∈ A , for 1 ≤ i ≤ p , | S ∗ ∩ N G ′ [ a i ] | ≥
1. Similarly, for every vertex b i ∈ B , for 1 ≤ i ≤ q , | S ∗ ∩ N G ′ [ b i ] | ≥ S ∗ is a dominating set of size at most k in G . (cid:3) Theorem 3.
SCDM is NP-complete for comb convex bipartite graphs.Proof.
It is known that SCDM is in NP. To prove the NP-hardness of SCDM forcomb convex bipartite graphs we reduce from Domination problem for bipartite graphs.Given an instance G = ( A , B , E ) of Domination problem for bipartite graphs, where A = a , a , . . . , a p } and B = { b , b , . . . , b q } , we construct an instance G ′ = ( A ′ , B ′ , E ′ ) ofSCDM, where A ′ = A ∪ { a ′ p + , a ′ p + , . . . , a ′ p } ∪{ a x , a y } , B ′ = B ∪ { b ′ p + , b ′ p + , . . . , b ′ p }∪ { b x } , and E ′ = E ∪ { ( a ′ i , b j ) : p + ≤ i ≤ p , ≤ j ≤ q } ∪ { ( a ′ i , b ′ i ) : p + ≤ i ≤ p } ∪ { ( a i , b x ) : 1 ≤ i ≤ p } ∪ { ( a ′ p + i , b x ) : 1 ≤ i ≤ p } ∪ { ( a x , b x ) , ( a y , b x ) } . It can be veri-fied that G ′ is a comb convex bipartite graph with its associated comb T = ( A ′ , F ), withbackbone { a ′ p + , a ′ p + , . . . , a ′ p , a x } and teeth { a , a , . . . , a p , a y } . It can be noted that theconstruction of graph G ′ can be done in polynomial time.Next we show that G has a dominating set of size at most k if and only if G ′ has aSCDS of size at most l = k + p +
3. Suppose D is a dominating set in G of size at most k . Then it can be easily verified that the set D ∪ { a ′ p + i , b ′ p + i : 1 ≤ i ≤ p } ∪ { a x , a y , b x } isa SCDS in G ′ of size at most k + p + S be a SCDS of size at most k + p + G ′ . Let A ∗ = { a ′ p + i :1 ≤ i ≤ p } and B ∗ = { b ′ p + i : 1 ≤ i ≤ p } . From Proposition 1, it is clear that | S ∩ A ∗ | = p , | S ∩ B ∗ | = p and | S ∩ { a x , a y , b x }| =
3. Suppose S ∗ = S ∩ V , then | S ∗ | ≤ k . Since S isa SCDS of G ′ , it can be easily verified that for every vertex v ∈ A ∪ B , N [ v ] ∩ S ∗ , ∅ .Therefore, S ∗ is a dominating set of size at most k . (cid:3) The following Vertex-Cover problem has been proved as NP-complete [16], which willbe used to show SCDM for chordal bipartite graphs as NP-complete.A vertex cover of an undirected graph G = ( V , E ) is a subset of vertices V ′ ⊆ V suchthat if edge ( u , v ) ∈ E , then either u ∈ V ′ or v ∈ V ′ or both. Vertex Cover Decision Problem (Vertex-Cover)
Instance:
A simple, undirected graph G and a positive integer k . Question:
Does there exist a vertex cover of size at most k in G ? Theorem 4.
SCDM is NP-complete for chordal bipartite graphs.Proof.
It is known that SCDM is in NP. To prove NP-hardness of SCDM for chordalbipartite graphs we reduce from Vertex-Cover. The reduction is similar to the construc-tion given in [19]. Given an instance G = ( V , E ) of Vertex-Cover, where | V | = n and | E | = m , we construct an instance G ′ = ( V ′ , E ′ ) of SCDM as follows.1. Replace each vertex i ∈ V by a component G i = ( V i , E i ) : • a i • b i • z i • d i • f i • x i • y i • c i • e i
2. Replace each edge ( i , j ) ∈ E by the following components G i j = ( V i j , E i j ) (Figure(a)) and G ji = ( V ji , E ji ) (Figure (b)) 5 • • • G ⇒ • x • y • c • e • b • z • d • f • a • x • y • c • e • b • z • d • f • a • b • z • d • f • a • x • y • c • e • b • z • d • f • a • x • y • c • e • p • r • q • s • r • p • s • q • p • r • q • s • r • p • s • q • p • r • q • s • r • p • s • q G ′ • u • t Figure 1: Example construction of graph G ′ from graph G • y j • r i j • s i j • x i • p i j • q i j (a) • y i • r ji • s ji • x j • p ji • q ji (b)Let X = { x i : i = , . . . , n } , Y = { y i : i = , . . . , n } , Z = { z i : i = , . . . , n } , K = X ∪ Y ∪ Z , A = { a i , b i , c i , d i , e i , f i : i = , . . . , n } , and B = { p i j , q i j , p ji , q ji , r i j , s i j , r ji , s ji :( i , j ) ∈ E } .3. Add two more additional vertices t and u such that V ′ = K ∪ A ∪ B ∪ { t , u } , E ′ = n S i = E i ∪ S ( i , j ) ∈ E ( E i j ∪ E ji ) ∪ { ( x i , y j ) , ( z i , y j ) : i = , . . . , n & j = , . . . , n } ∪{ ( x i , u ) , ( z i , u ), ( y i , t ) : i = , . . . , n } ∪ { ( t , u ) } .Since V ′ can be partitioned into two independent sets X ∪ Z ∪ { a i , c i , f i : i = , . . . , n } ∪{ q i j , q ji , r i j , r ji : ( i , j ) ∈ E }∪{ t } and Y ∪{ b i , d i , e i : i = , . . . , n }∪{ p i j , p ji , s i j , s ji : ( i , j ) ∈ E } ∪ { u } , the constructed graph G ′ is a bipartite graph.Let C be a cycle in G ′ of length greater than 4 . If C is a cycle within a component G i for some i , then clearly it contains y i . Otherwise, if C is a cycle formed with ver-tices from more than one G i component then it contains either edge ( x k , y l ) or ( z k , y l ).6herefore, each cycle of length greater than 4 contains at least one vertex y i ∈ Y . If C contains exactly one y i ∈ Y , (i) if C = G i then ( y i , z i ) is a chord, (ii) if C contains u then( u , z j ) is a chord, and (iii) if C contains t then ( y i , z j ) is a chord. If C contains at leasttwo vertices y i , y j from Y and (i) if C contains c i or c j then ( y i , z i ) or ( y j , z j ) is a chord,(ii) if C contains r i j or r ji then ( y i , c j ) is a chord, (iii) since vertices u and t are adjacentto every vertex v ′ ∈ X ∪ Z and u ′ ∈ Y respectively, if C contains t or u then there exists achord. Therefore, G ′ is a chordal bipartite graph and can be constructed in polynomialtime. An example construction of graph G ′ from graph G is illustrated in Figure 1.We show that G has a vertex cover of size at most k if and only if G ′ has a SCDS ofsize at most 7 n + m + k + . Let VC be a vertex cover of G of size at most k . Let S = { a i , b i , c i , d i , e i , f i : i ∈ V } ∪ { p i j , q i j , r i j , s i j , p ji , q ji , r ji , s ji : ( i , j ) ∈ E } ∪ { x i , y i : i ∈ VC } ∪ { z i : i < VC } ∪ { t , u } . It can be verified that S forms a SCDS of G ′ and | S | = n + m + k + ( n − k ) + = n + m + k + . Conversely, suppose S ′ is a SCDS of size at most 7 n + m + k + . Claim 1.
If x i ∈ S ′ then without loss of generality, y i ∈ S ′ and vice versa.Proof of claim. Let x i ∈ S ′ . Since S ′ is a CDS, then it is true that either y i ∈ S ′ or z i ∈ S ′ . Then, take without loss of generality, y i ∈ S ′ . Analogously, if y i ∈ S ′ , theneither x i ∈ S ′ or z i ∈ S ′ . Then, take without loss of generality, x i ∈ S ′ . (cid:3) Claim 2.
If S ′ is a SCDS of G ′ with | S ′ ∩ { t , u }| < then there exists a SCDS of G ′ withthe same size and | S ′ ∩ { t , u }| = .Proof of claim. Since X ∪ Z ∪ { t } and Y ∪ { u } forms a complete bipartite subgraph in G ′ , if S ′ is a SCDS of G ′ and t , u < S ′ then there exists two vertices v ∈ S ′ ∩ Y , v ∈ S ′ ∩ ( X ∪ Z ) such that ( S ′ \ { v , v } ) ∪ { t , u } is also a SCDS of G ′ . With the similarargument, if t < S ′ (or u < S ′ ) then there exists a vertex v ∈ S ′ ∩ Y (or v ∈ S ′ ∩ X )such that ( S ′ \ { v } ) ∪ { t } (or ( S ′ \ { v } ) ∪ { u } ) is a SCDS of G ′ . Hence the claim. (cid:3)
Let S = { a i , b i , c i , d i , e i , f i : 1 ≤ i ≤ n } and S = { p i j , q i j , r i j , s i j , p ji , q ji , r ji , s ji :( i , j ) ∈ E } . From Proposition 1, it is true that S ⊂ S ′ , and also S ⊂ S ′ . Let S ∗ = S ′ \ ( S ∪ S ∪ { t , u } ) . Clearly, | S ∗ | ≤ n + k . Let | S ∗ ∩ X | = k ′ . From claim 1, clearly | S ∗ ∩ ( X ∪ Y ) | = k ′ . Since S ′ is also a CDS of G ′ , | S ∗ ∩ Z | = n − k ′ . Thus,2 k ′ + ( n − k ′ ) ≤ n + kk ′ ≤ k (1) Claim 3.
If VC = { i : x i , y i ∈ S ′ } , then VC forms a vertex cover in G . Proof of claim.
Let ( i , j ) ∈ E . From the construction of G , it can be observed that thereis no path from p i j to b i without x i or y j . Since S ′ is connected, it should contain either x i or y j for each G i j . Similar argument can be made for each G ji . Therefore, for each( i , j ) ∈ E either x i , y i ∈ S ′ or x j , y j ∈ S ′ . Hence, VC is a vertex cover in G . (cid:3) Therefore, from above claim and equation (1), clearly there exists a vertex cover ofsize at most k . (cid:3) y • y • y • x • x • x (a) • y • y • y • y • x • x • x • x (b) • y • y • y • y • x • x • x • x (c)Figure 2: SCDS in Chain graphs In this section, we propose a method to compute a minimum SCDS of a chain graph inlinear time. A bipartite graph G = ( X , Y , E ) is called a chain graph if the neighborhoodsof the vertices of X form a chain , that is, the vertices of X can be linearly orderedsay, x , x , . . . , x p , such that N ( x ) ⊆ N ( x ) ⊆ . . . ⊆ N ( x p ) . If a bipartite graph G = ( X , Y , E ) is a chain graph, then the neighborhoods of the vertices of Y also form a chain.An ordering α = ( x , x , . . . , x p , y , y , . . . , y q ) of X ∪ Y is called a chain ordering if N G ( x ) ⊆ N G ( x ) ⊆ . . . ⊆ N G ( x p ) and N G ( y ) ⊇ N G ( y ) ⊇ . . . ⊇ N G ( y q ). Every chaingraph admits a chain ordering [26]. Theorem 5.
SCDM is linear time solvable for chain graphs.Proof.
Let G = ( X , Y , E ) be a chain graph with chain ordering { x , x , . . . , x p , y , y , . . . , y q } .If p = q = G is a complete bipartite graph and clearly, γ sc ( G ) = | X ∪ Y | .Otherwise, Let S = { y , y , x p − , x p } ∪ P , where P contains all the pendant vertices of G . It can be observed that for every vertex u ∈ V \ S there exists a vertex v ∈ S suchthat ( S \ { v } ) ∪ { u } is a CDS of G . Hence, S is a SCDS of G and γ sc ( G ) ≤ | S | . Let S ′ be any SCDS of G , then we show that | S ′ | ≥ | S | . Note that if X ∩ P , ∅ ( Y ∩ P , ∅ ) then y ( x p ) is a support vertex. It is known that every SCDS contains allthe pendant and support vertices of G . If P , ∅ (Figure 2(a) & (b)) then clearly | S ′ | ≥| S | . Otherwise, if | ( S ′ ∩ Y ) | < u ∈ X \ S ′ for which there isno vertex v ∈ S ′ such that ( S ′ \ { v } ) ∪ { u } is a CDS of G . Thus, | ( S ′ ∩ Y ) | ≥ | ( S ′ ∩ X ) | ≥ . Hence, | S ′ | ≥ | S | . In a chain graph G = ( X , Y , E ), a chain ordering and the set P of all pendant verticescan be computed in linear time [23]. Therefore, SCDM in chain graphs can be solvedin linear time. (cid:3) Let G be a graph, T be a tree and v be a family of vertex sets V t ⊆ V ( G ) indexed bythe vertices t of T . The pair ( T , v ) is called a tree-decomposition of G if it satisfies thefollowing three conditions: (i) V ( G ) = S t ∈ V ( T ) V t , (ii) for every edge e ∈ E ( G ) thereexists a t ∈ V ( T ) such that both ends of e lie in V t , (iii) V t ∩ V t ⊆ V t whenever t , t ,8 ∈ V ( T ) and t is on the path in T from t to t . The width of ( T , v ) is the number max {| V t | − t ∈ T } , and the tree-width tw ( G ) of G is the minimum width of anytree-decomposition of G . By Courcelle’s Thoerem, it is well known that every graphproblem that can be described by counting monadic second-order logic (CMSOL) canbe solved in linear-time in graphs of bounded tree-width, given a tree decompositionas input [8]. We show that SCDM problem can be expressed in CMSOL. Theorem 6 ( Courcelle’s Theorem ) . ([8]) Let P be a graph property expressible inCMSOL and let k be a constant. Then, for any graph G of tree-width at most k, it canbe checked in linear-time whether G has property P. Theorem 7.
Given a graph G and a positive integer k, SCDM can be expressed inCMSOL.Proof.
First, we present the CMSOL formula which expresses that the graph G has adominating set of size at most k . Dominating ( S ) = ( | S | ≤ k ) ∧ ( ∀ p )(( ∃ q )( q ∈ S ∧ ad j ( p , q ))) ∨ ( p ∈ S ) where ad j ( p , q ) is the binary adjacency relation which holds if and only if, p , q are twoadjacent 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 is at most k . For a set S ⊆ V , the induced subgraph G [ S ] is disconnected if and only if the set S can be partitionedinto two sets S and S such that there is no edge between a vertex in S and a vertexin S . The CMSOL formula to express that the induced subgraph G [ S ] is connected asfollows. Connected ( S ) = ¬ ( ∃ S , S ⊆ S , ¬ ( ∃ e ∈ E , ∃ u ∈ S , ∃ v ∈ S \ S , ( inc ( u , e ) ∧ inc ( v , e )))) where inc ( v , e ) is the binary incidence relation which hold if and only if edge e isincident to vertex v in G . Now, by using the above two CMSOL formulas we canexpress SCDM in CMSOL formula as follows.
S CDM ( S ) = Dominating ( S ) ∧ Connected ( S ) ∧ ( ∀ x )(( x ∈ S ) ∨ (( ∃ y )( y ∈ S ∧ Dominating (( S \ { y } ) ∪ { x } ) ∧ Connected (( S \ { y } ) ∪ { x } )))) Therefore, SCDM can be expressed in CMSOL. (cid:3)
Now, the following result is immediate from Theorems 6 and 7.
Theorem 8.
SCDM can be solvable in linear time for bounded tree-width graphs.
In this section, we obtain upper and lower bounds on the approximation ratio of theMSCDS problem. We also show that the MSCDS problem is APX-complete for graphswith maximum degree 4. 9 .1 Approximation Algorithm
Here, we propose a ∆ ( G ) + Theorem 9. ([7]) The MINIMUM DOMINATION problem in a graph with maximumdegree ∆ ( G ) can be approximated with an approximation ratio of + ln( ∆ ( G ) + . Theorem 10. ([11]) The MINIMUM CONNECTED DOMINATION problem in a graphwith maximum degree ∆ ( G ) can be approximated with an approximation ratio of + ln ∆ ( G ) . By theorems 9 and 10, let us consider APPROX-DOM-SET and APPROX-CDSare the approximation algorithms to approximate the solutions for MINIMUM DOM-INATION and MINIMUM CONNECTED DOMINATION with approximation ratiosof 1 + ln( ∆ ( G ) +
1) and 3 + ln ∆ ( G ) respectively.Now, we propose an algorithm APPROX-SCDS to produce an approximate solu-tion for the MSCDS problem. In APPROX-SCDS, first we compute CDS D c of a givengraph G using APPROX-CDS. Next, we obtain the induced subgraph G ′ from V \ D c .By using APPROX-DOM-SET, we compute dominating set D of G ′ . Let D sc = D c ∪ D . It can be easily observed that for every vertex u ∈ V \ D sc there exists a vertex v ∈ D such that ( D sc \ { v } ) ∪ { u } is a CDS of G . Therefore, D sc is a SCDS of G . Algorithm 1
APPROX-SCDS( G ) Input:
A simple and undirected bipartite graph G Output:
A SCDS D sc of G . D c ← APPROX-CDS ( G ) Let G ′ = G [ V \ D c ] D ← APPROX-DOM-SET ( G ′ ) D sc ← D c ∪ D return D sc . Theorem 11.
The MSCDS problem in a graph G with maximum degree ∆ ( G ) can beapproximated with an approximation ratio of ( ∆ ( G ) + . Proof.
To prove the theorem, we show that SCDS produced by our algorithm APPROX-SCDS, D sc , is of size at most ( ∆ ( G ) +
1) times of γ sc ( G ), i.e., | D sc | ≤ ( ∆ ( G ) + γ sc ( G )From the algorithm, | D sc | = | D c ∪ D | = | D c | + | D | ≤ n ≤ ( ∆ ( G ) + γ ( G ) ≤ ( ∆ ( G ) + γ sc ( G )10 Since the MSCDS problem in a graph with maximum degree ∆ ( G ) admits an ap-proximation algorithm that achieves the approximation ratio of ( ∆ ( G ) + Corollary 1.
The MSCDS problem is in the class of APX when the maximum degree ∆ ( G ) is fixed. To obtain a lower bound, we provide an approximation preserving reduction from theMINIMUM DOMINATION problem, which has the following lower bound.
Theorem 12. [5] For a graph G = ( V , E ) , the MINIMUM DOMINATION problem can-not be approximated within (1 − ǫ ) ln n for any ǫ > unless NP ⊆ DTIME ( n O (log log n ) ) ,where n = | V | . The above result holds in bipartite and split graphs as well [5].
Theorem 13.
For a graph G = ( V , E ) , the MSCDS problem cannot be approximatedwithin (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 approximation preserv-ing reduction. Let G = ( V , E ), where V = { v , v , . . . , v n } be an instance of the MINI-MUM DOMINATION problem. From this we construct an instance G ′ = ( V ′ , E ′ ) ofMSCDS, where V ′ = V ∪ { w , z } , and E ′ = E ∪ { ( v i , w ) : v i ∈ V } ∪ { ( w , z ) } .Let D ∗ be a minimum dominating set of a graph G and S ∗ be a minimum SCDS ofa graph G ′ . It can be observed from the reduction that by using any dominating set of G , a SCDS of G ′ can be formed by adding w and z vertices to it. Hence | S ∗ | ≤ | D ∗ | + . Let algorithm A be a polynomial time approximation algorithm to solve the MSCDSproblem 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 the following algorithm,DOM-SET-APPROX to find a dominating set of a given graph G . Algorithm 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 SCDS S of G ′ by using algorithm A D ← S ∩ V end if return D . D is a minimum dominating set of size at most k , then it is optimal. Next, we analyzethe case where D is not a minimum dominating set of size at most k . Let S ∗ be a minimum SCDS of G ′ , then | S ∗ | ≥ k . Given a graph G , DOM-SET-APPROX computes a dominating set of size | D | ≤ | S | ≤ α | S ∗ | ≤ α ( | D ∗ | + = α (1 + / | D ∗ | ) | D ∗ | ≤ α (1 + / k ) | D ∗ | . Therefore, DOM-SET-APPROX approximatesa dominating set within a ratio α (1 + / k ) . If 2 / k < ǫ/ , then the approximation ratio α (1 + / 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 approximatedwithin a ratio of (1 − ǫ ′ ) ln n , then NP ⊆ DT I ME ( n O (log log n ) ). Similarly, if the MSCDSproblem can be approximated within a ratio of (1 − ǫ ) ln n , then NP ⊆ DT I ME ( n O (log log n ) ).For large values of n , ln n ≅ ln( n + G ′ = ( V ′ , E ′ ) , where | V ′ | = | V | + , MSCDS problem cannot be approximated within a ratio of (1 − ǫ ) ln | V ′ | unless NP ⊆ DT I ME ( | V ′ | O (log log | V ′ | ) ) . (cid:3) Theorem 14.
For a bipartite graph G = ( X , Y , E ) , the MSCDS problem cannot beapproximated within (1 − ǫ ) ln n for any ǫ > unless NP ⊆ DTIME ( n O (log log n ) ) , wheren = | X ∪ Y | .Proof. In order to prove the theorem, we propose the following approximation pre-serving reduction. Consider G = ( X , Y , E ), where X = { x , x , . . . , x p } and Y = { y , y , . . . , y q } be an instance of the MINIMUM DOMINATION problem. From thiswe construct an instance G ′ = ( X ′ , Y ′ , E ′ ) of MSCDS, where X ′ = X ∪{ w , z } , Y ′ = Y ∪{ z , w } and E ′ = E ∪ { ( x i , z ) : x i ∈ X } ∪ { ( y i , w ) : y i ∈ Y } ∪ { ( w , w ) , ( z , z ) , ( w , z ) } .An example construction of graph G ′ from a bipartite graph G = ( X , Y , E ) with X = { x , x , x , x , x } , and Y = { y , y , y , y , y } is illustrated in Figure 3.Let D ∗ be a minimum dominating set of a graph G and S ∗ be a minimum SCDSof a graph G ′ . It can be observed from the reduction that by using any dominating setof G , a SCDS of G ′ can be formed by adding { w , w , z , z } vertices to it. Hence, | S ∗ | ≤ | D ∗ | + . The rest of the proof is similar to the proof of theorem 13. (cid:3)
In this subsection, we prove that the MSCDS problem is APX-complete for graphswith maximum degree 4. This can be proved using an L-reduction, which is defined asfollows.
Definition 1. (L-reduction)
Given two NP optimization problems F and G and a poly-nomial time transformation f from instances of F to instances of G, one can say that fis an L-reduction if there exists positive constants α and β such that for every instancex of F the following conditions are satisfied.1. 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 inpolynomial 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 | . x • x • x • x • x • y • y • y • y • y • z • w • z • w Figure 3: Example construction of a graph G ′ Here, opt F ( x ) represents the size of an optimal solution for an instance x of an NPoptimization 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-completeproblem to π .By theorem 11, it is known that the MSCDS problem can be approximated within aconstant factor for graphs with maximum degree 4. Thus, this problem is in APX forgraphs with maximum degree 4. To show APX-hardness of MSCDS, we give an L-reduction from MINIMUM DOMINATING SET problem in graphs with maximumdegree 3 (DOM-3) which has been proved as APX-complete [1]. Theorem 15.
The MSCDS problem is APX-complete for graphs with maximum degree . Proof.
It is known that MSCDS 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 MSCDS asfollows. Let X = { x , x , . . . , x n } and Y = { y , y , . . . , y n } . In graph G ′ , V ′ = V ∪ X ∪ Y and E ′ = E ∪ { ( v i , x i ) , ( x i , y i ) : 1 ≤ i ≤ n } ∪ { ( x i , x i + ) : 1 ≤ i ≤ n − } . Note that G ′ isa graph with maximum degree 4. An example construction of a graph G ′ from a graph G is shown in Figure 4. Claim 4.
If D ∗ is a minimum dominating set of G and S ∗ is a minimum SCDS of G ′ then | S ∗ | = | D ∗ | + n , where n = | V | . Proof of claim.
Suppose D ∗ is a minimum dominating set of G , then D ∗ ∪ X ∪ Y isa SCDS of G ′ . Further, if S ∗ is a minimum SCDS of G ′ , then it is clear that | S ∗ | ≤| D ∗ | + n . • v • x • y • v • x • y • x • y • v • x • y • v Figure 4: Construction of G ′ from G Next, we show that | S ∗ | ≥ | D ∗ | + n . Let S be any SCDS of G ′ . From Proposition1, it is clear that X ∪ Y ⊂ S , and no vertex w ∈ X ∪ Y is an S - defender . Therefore,for every vertex u < S there exists a vertex v ∈ S ∩ V such that ( S \ { v } ) ∪ { u } is aCDS of G ′ . Hence D = S ∩ V is a dominating set of G and | D | ≥ | D ∗ | which implies | S | ≥ | D ∗ | + n . Since | S | ≥ | S ∗ | , it is clear that | S ∗ | ≥ | D ∗ | + n . (cid:3) Let D ∗ and S ∗ be a minimum dominating set and minimum SCDS of G and G ′ respectively. It is known that for any graph H with maximum degree ∆ ( H ), γ ( H ) ≥ n ∆ ( H ) + , where n = | V ( H ) | . Thus, | D ∗ | ≥ n . From above claim it is evident that, | S ∗ | = | D ∗ | + n ≤ | D ∗ | + | D ∗ | = | D ∗ | . Now, consider a SCDS S of G ′ . Clearly, the set D = S ∩ V is a dominating set of G . Therefore, | D | ≤ | S | − n . Hence, | D | − | D ∗ | ≤ | S | − n − | D ∗ | = | S | − | S ∗ | . This provesthat there is an L-reduction with α = β = . (cid:3) ff erence in domination and secure con-nected domination Although secure connected domination is one of the several variants of dominationproblem, however they di ff er in computational complexity. In particular, there existgraph classes for which the first problem is polynomial-time solvable whereas the sec-ond problem is NP-complete and vice versa. Similar study has been made betweendomination and other domination parameters in [14, 20, 21].The DOMINATION problem is linear time solvable for doubly chordal graphs [3],but the SCDM problem is NP-complete for this class of graphs which is proved insection 2.1. Now, we construct a class of graphs in which the MSCDS problem can besolved trivially, whereas the DOMINATION problem is NP-complete. Definition 2. (GC graph)
A graph is GC graph if it can be constructed from a con-nected graph G = ( V , E ) where | V | = n , in the following way:1. Create n complete graphs each with vertices, such that i th complete graph contains v • x • a • b • c • v • x • a • b • c • x • a • b • c • v • x • a • c • b • v G Figure 5: GC graph construction vertices { a i , b i , c i } .2. Create n vertices, { x , x , . . . , x n } .
3. Add edges { ( x i , v i ) , ( x i , a i ) : v i ∈ V } . Theorem 16.
If G ′ is a GC graph obtained from a graph G = ( V , E ) ( | V | = n ) , then γ sc ( G ′ ) = n.Proof. Let G ′ = ( V ′ , E ′ ) be a GC graph. An example construction of GC graph isillustrated in Figure 5. Let S = V ∪ { x , x , . . . , x n } ∪ { a i , b i : 1 ≤ i ≤ n } . It can beobserved that S is a SCDS of G ′ of size 4 n and hence γ sc ( G ′ ) ≤ n . Let S be any SCDS in G ′ . It is known that every SCDS of a graph G is also a CDSof G and every CDS should contain all the cut-vertices of G . Thus, it can be easilyobserved that for 1 ≤ i ≤ n , the vertices v i , x i and a i are cut-vertices in G ′ and thesevertices should be included in every SCDS of G ′ . Therefore, | S | ≥ n . It can also benoted that these cannot defend any other vertex in G ′ . Therefore, either b i or c i , for each i , where 1 ≤ i ≤ n should be included in every SCDS of G ′ , and hence, | S | ≥ n . (cid:3) Lemma 1.
Let G ′ be a GC graph constructed from a graph G = ( V , E ) . Then G has adominating set of size at most k if and only if G ′ has a dominating set of size at mostk + n . Proof.
Let A contain the degree 3 vertex from each copy of K . Suppose D is a domi-nating set of G of size at most k , then it is clear that D ∪ A is a dominating set of G ′ ofsize at most k + n . Conversely, suppose D ′ is a dominating set of G ′ of size k + n . Then at least onevertex from each k must be included in D ′ . Let D ′′ be the set formed by replacing all x i ’s in D ′ with corresponding v i ’s. Clearly, D ′′ is a dominating set of size at most k in G . (cid:3) The following result is well known for the DOMINATION problem.
Theorem 17. ([10]) The DOMINATION problem is NP-complete for general graphs. heorem 18. The DOMINATION problem is NP-complete for GC graphs.Proof.
The proof directly follows from above theorem and lemma 1. (cid:3)
It is identified that the two problems, DOMINATION and SCDM are not equivalentin aspects of computational complexity. For example, when the input graph is eitherdoubly chordal or a GC graph then complexities di ff er. Thus, there is a scope to studyeach of these problems on its own for particular graph classes. Acknowledgement
The authors are grateful to the anonymous reviewers for their valuable comments andsuggestions, which result in the present version of the paper.
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