Algorithmic Complexity of Secure Connected Domination in Graphs
aa r X i v : . [ c s . D M ] F e b Algorithmic Complexity of Secure Connected Domination in Graphs
Jakkepalli Pavan Kumar, P. Venkata Subba Reddy
Department of Computer Science and EngineeringNational Institute of Technology Warangal, Warangal, Telangana, India
S. Arumugam
Director, n-CARDMATH,Kalasalingam Academy of Research and EducationAnand Nagar, Krishnankoil, Tamilnadu, India.
Abstract
Let G = ( V, E ) be a simple, undirected and connected graph. A connected (total) dominating set S ⊆ V is a secure connected (total) dominating set of G , if for each u ∈ V \ S , there exists v ∈ S suchthat uv ∈ E and ( S \{ v } ) ∪{ u } is a connected (total) dominating set of G . The minimum cardinalityof a secure connected (total) dominating set of G denoted by γ sc ( G )( γ st ( G )), is called the secureconnected (total) domination number of G . In this paper, we show that the decision problemscorresponding to secure connected domination number and secure total domination number areNP-complete even when restricted to split graphs or bipartite graphs. The NP-complete reductionsalso show that these problems are w[2]-hard. We also prove that the secure connected dominationproblem is linear time solvable in block graphs and threshold graphs. Keywords:
Domination, Secure domination, Secure connected domination, W[2]-hard.
1. Introduction
Let G ( V, E ) be a simple, undirected and connected graph. For graph theoretic terminology werefer to [9]. For a vertex v ∈ V , the open neighborhood of v in G is N G ( v )= { u ∈ V : uv ∈ E } , the closed neighborhood of v is defined as N G [ v ] = N G ( v ) ∪ { v } . If S ⊆ V , then the open neighborhoodof 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 respect to S if N [ w ] ∩ S = { v } . If further w ∈ V \ S , then w is called an external private neighbor (epn) of v .A subset S of V is a dominating set (DS) in G if for every u ∈ V \ S , there exists v ∈ S suchthat uv ∈ E . The domination number of G is the minimum cardinality of a DS in G and is denotedby γ ( G ). A set S ⊆ V is said to be a secure dominating set (SDS) in G if for every u ∈ V \ S there exists v ∈ S such that uv ∈ E and ( S \ { v } ) ∪ { u } is a dominating set of G . We say that v S - defends u or u is defended by v . The minimum cardinality of a SDS in G is called the secure Email addresses: [email protected] (Jakkepalli Pavan Kumar), [email protected] (P. Venkata Subba Reddy), [email protected] (S. Arumugam)
Preprint submitted to Elsevier February 6, 2020 omination number of G and is denoted by γ s ( G ). A dominating set S is said to be a connecteddominating set (CDS), if the induced subgraph G [ S ] is connected. A CDS S is said to be a secureconnected dominating set (SCDS) in G if for each u ∈ V \ S , there exists v ∈ S such that uv ∈ E and ( S \ { v } ) ∪ { u } is a CDS in G . The minimum cardinality of a SCDS in G is called the secureconnected domination number of G and is denoted by γ sc ( G ). A dominating set S is said to be a total dominating set (TDS), if the induced subgraph G [ S ] has no isolated vertices. A TDS S is saidto be a secure total dominating set (STDS) of G , if for each u ∈ V \ S , there exists v ∈ S such that uv ∈ E and ( S \ { v } ) ∪ { u } is a TDS in G . The minimum cardinality of a STDS in G is called the secure total domination number of G and is denoted by γ st ( G ). We need the following theorems. Theorem 1. ([2]) Let G be a connected graph of order n . Then γ sc ( G ) = 1 if and only if G = K n . Theorem 2. ([2]) Let G be a connected graph of order n ≥ . Let L ( G ) and S ( G ) be the set ofpendant and support vertices of G respectively. Let X be a secure connected dominating set of G .Then (i) L ( G ) ⊆ X and S ( G ) ⊆ X (ii) No vertex in L ( G ) ∪ S ( G ) is an X-defender. Proposition 1. ([3]) Let S be a CDS in G . Then S is a SCDS in G if and only if the followingconditions are satisfied.(i) epn ( v, S ) = ∅ for all v ∈ S .(ii) For every u ∈ V \ S , there exists v ∈ S ∩ N G ( u ) such that V ( C ) ∩ N G ( u ) = ∅ for everycomponent C of G [ S \ { v } ] . Proposition 2. ([2]) Let G be a non-complete connected graph and let S be a secure connecteddominating set in G . Then the set S \ { v } is a dominating set for every v ∈ S . In particular, γ ( G ) ≤ γ sc ( G ) .
2. Main Results
We first determine the value of γ sc ( G ) for two families of graphs. Theorem 3.
Let W n = v + C n be the wheel of order n + 1 where n ≥ . Let G be the graphobtained from W n +1 by subdividing all the edges of C n . Then γ sc ( G ) = n + 1 . Proof.
Let V ( G ) = { v , v , . . . , v n +1 } , d ( v n +1 ) = n , d ( v i ) = ( i is even3 otherwise and N ( v i ) = { v i − , v i +1 } if i is even. Then S = { v i : i is odd } is a SCDS of G. Hence γ sc ( G ) ≤ n + 1.Now let D be any γ sc -set of G . If v n +1 / ∈ D or if v n +1 ∈ D and defends a vertex v i ,then we get a connected dominating set D of G such that | D | = | D | and v n +1 / ∈ D . Hence | D | = | D | ≥ n −
2, which is a contradiction. Thus v n +1 ∈ D and v n +1 does not defend anyother vertex. Now let v i ∈ D for some i where i is even. Since G [ D ] is connected, one of v i − or v i +1 is in D. Also if v i / ∈ D for all even i, then v i ∈ D for all odd i. Hence γ sc ( G ) = | D | ≥ n + 1 . Theorem 4.
For the Book graph B n = K ,n (cid:3) K , we have γ sc ( B n ) = n + 2 .Proof. Let S and S be the two copies of K ,n in B n . Let V ( S ) = { v , v , . . . , v n +1 } and V ( S ) = { w , w , . . . , w n +1 } . Let v and w be the central vertices of S , S respectively. Let v i w i ∈ E ( B n ).Clearly V ( S ) ∪ { w } is an SCDS of B n . Hence γ sc ( B n ) ≤ n + 2 . D be any γ sc -set of B n . Since D is connected, either v or w is in D. If w ∈ D and v / ∈ D , then { w , w , . . . , w n +1 , v , v , . . . , v n +1 } ⊆ D. Thus, | D | ≥ n + 1 which is a contradiction.Hence v , w ∈ D. Now if both w i and v i are not in D for some i ≥ , then G [( D \ { w } ) ∪ { w i } ] and G ( D \{ v } ) ∪{ v i } ] are disconnected. Hence | D ∩{ w i , v i }| ≥ i ≥ γ sc ( B n ) = | D | ≥ n +2 . Thus, γ sc ( B n ) = n + 2. Theorem 5.
Let G = P n (cid:3) P where n ≥ . Then γ sc ( G ) = n + ⌈ n ⌉ . Proof.
Let P = ( v , v , . . . , v n ) and Q = ( w , w , . . . , w n ) be two copies of P n in G such that v i w i ∈ E ( G ) . Let V = { v , v , . . . , v n } and V = { w , w , . . . , w n } . Then S = V ∪ { w i : i ≡ mod } is aSCDS of G . Hence γ sc ( G ) ≤ n + ⌈ n ⌉ . Let D be any γ sc -set of G. If v i , w i / ∈ D for some i, where 2 ≤ i ≤ n −
1, then G [ D ] isdisconnected, which is a contradiction. Hence at least one of v i , w i is in D , where 2 ≤ i ≤ n − . Ifboth v and w are not in D , then G [( D \ { v } ) ∪ { v } ] and G [( D \ { w } ) ∪ { w } ] are disconnected,which is a contradiction. Hence v or w is in D. Similarly, w n or v n is in D. We now claim that D ∩ V is a dominating set of P. Suppose there exists a vertex v i such that v i is not dominatedby D ∩ V . Then w i ∈ D and G [( D \ { w i } ) ∪ { v i } ] is disconnected, which is a contradiction.Hence D ∩ V is a dominating set of P. Similarly D ∩ V is a dominating set of Q. Now suppose D ∩ V ( V and D ∩ V ( V . If three consecutive vertices of P say, v i , v i +1 , v i +2 are not in D , then w i , w i +1 , w i +2 ∈ D. However, G [( D \ { w i +1 } ) ∪ { v i +1 } ] is disconnected, which is a contradiction.Now suppose v i , v i +1 / ∈ D . Then v i − , v i +2 , w i , w i +1 ∈ D. Now since G [ D ] is connected, it followsthat w i − , w i +2 ∈ D. Hence ( D \ { w i , w i +1 } ) ∪ { v i , v i +1 } is also a SCDS of G. Thus by repeatingthe above process we get a SCDS of
G, D such that | D | = | D | , D ∩ V = V and D ∩ V is adominating set of Q. Thus, | D | = | D | ≥ n + ⌈ n ⌉ . Therefore, γ sc ( G ) = n + ⌈ n ⌉ . We now proceed to present results on algorithmic aspects such as NP-comppleteness and lineartime algorithm for some classes of graphs.
Secure Connected Domination Problem ( SCDM ) Instance : A connected graph G and a positive integer l . Question : Does there exist a SCDS of size at most l in G ?The proof is by reduction from the Domination problem (DM), which is NP-complete [5]. Domination Problem ( DM ) Instance : A graph G and a positive integer k . Question : Does there exist a DS of size at most k in G ? Theorem 6.
SCDM is NP-complete.Proof.
It can be easily verified that SCDM is in NP. Now let G = ( V, E ) be a graph and let k bea positive integer. Let G ∗ be the graph with V ( G ∗ ) = V ∪ { x } and E ( G ∗ ) = E ∪ { ( u, x ) : u ∈ V } and let l = k + 1. Clearly, G ∗ can be constructed from G in polynomial time.Now if D is a dominating set of G with | D | ≤ k , then S = D ∪ { x } is an SCDS of G ∗ .Conversely, let S ∗ be an SCDS of G ∗ with | S ∗ | ≤ k + 1 . If x ∈ S , then it follows from Proposition 1that epn ( x, S ) = ∅ . Therefore, every vertex u ∈ V ( G ∗ ) \ S is adjacent to a vertex in S \ { x } . Hence S \ { x } is a DS of size at most k in G . If x / ∈ S , Proposition 2, the set S \ { v } , for any v ∈ S , is aDS of size at most k in G . 3ext, we define the decision version of total domination and secure total domination problems asfollows. Total Domination Problem ( TDM ) Instance : A simple, undirected graph G without isolated vertices and a positive integer r . Question : Does there exist a TDS of size at most r in G ? Secure Total Domination Problem ( STDM ) Instance : A simple, undirected and connected graph G and a positive integer m . Question : Does there exist a STDS of size at most m in G ? Theorem 7.
STDM is NP-complete.Proof.
It is clear that STDM is in NP. The reduction given in the proof of Theorem 6 shows thatSTDM is NP-complete.We now give NP-completeness results even when restricted to bipartite graphs or split graphs.We formulate the SCDM for bipartite graphs as follows.
Secure Connected Domination Problem for Bipartite Graphs ( SCDB ) Instance:
A connected bipartite graph G = ( V , V , E ) and a positive integer r . Question:
Does there exist a SCDS of size at most r in G ? Theorem 8.
SCDB is NP-complete.Proof.
It can be seen that SCDB is in NP. We transform an instance of SCDM problem to aninstance of SCDB as follows. Given a graph G , we construct a graph G ∗ ( V , V , E ) where V ( G ∗ ) = V ∪ { p, q } , V ( G ∗ ) = V ′ ( G ) ∪ { x, y } , here V ′ ( G ) is another copy of V such that if u and v are twovertices in V then the corresponding vertices in V ′ ( G ) are labeled as u ′ and v ′ , and E ( G ∗ ) consistsof (i) edges uv ′ and u ′ v for each edge uv ∈ E ; (ii) edges of the form uu ′ for each vertex u ∈ V ; and(iii) edges of the form ux and uy for every vertex u ∈ V ( G ∗ ). Clearly G ∗ is a bipartite graph andcan be constructed from G in polynomial time.Next, we show that G has a SCDS of size at most r if and only if G ∗ has a SCDS of size atmost r + 2. If S is a SCDS of G with | S | ≤ r, then it can be easily verified that S ∗ = S ∪ { x, y } isa SCDS of G ∗ with | S ∗ | ≤ r + 2 . Conversely, let S ∗ be an SCDS of G ∗ and | S ∗ | ≤ r + 2. Since x and y are the only verticesin S ∗ which defend p and q , it follows that at least one of them must be in S ∗ . If x ∈ S ∗ and y / ∈ S ∗ , then G ∗ [( S ∗ \ { x } ) ∪ { p } ] is disconnected, which is a contradiction. Hence x, y ∈ S ∗ . Let S ′ = S ∗ \ { x, y, p, q } and S ′′ = ( S ′ ∪ { v : v ′ ∈ S ′ ∩ V ′ ( G ) } ) \ { v ′ : v ′ ∈ S ′ ∩ V ′ ( G ) } . Clearly S ′′ formsa SCDS of size at most r in G . Theorem 9.
STDM is NP-complete for bipartite graphs.Proof.
It is clear that STDM for bipartite graphs is in NP. The reduction given in the proof ofTheorem 8 shows that STDM is NP-complete for bipartite graphs.Since the Domination problem is w[2]-complete for bipartite graphs [8] and the reductions in The-orem 8 and Theorem 9 are in the function of the parameter l, the following two corollaries areimmediate. Corollary 1.
SCDM is w[2]-hard in bipartite graphs. b cd e (a) Graph G abcdepq a ′ b ′ c ′ d ′ e ′ xy (b) Graph G ∗ Figure 1: Construction of G ∗ from G Corollary 2.
STDM is w[2]-hard in bipartite graphs.
It has been shown that the DM and the TDM as NP-complete even when restricted to split graphs[1].
Theorem 10.
SCDM is NP-complete for split graphs.Proof.
It is known that SCDM is a member of NP. We reduce DM for split graphs to SCDM for splitgraphs. Given a split graph G whose vertex set is partitioned into a clique Q and an independentset I , we construct a split graph G ∗ with a clique Q ∗ and an independent set I ∗ as follows: V ( G ∗ ) = V ∪ { x, y } , and E ( G ∗ ) = E ∪ { xu : u ∈ V } ∪ { xy } .Note that G ∗ is a split graph, where Q ∗ = Q ∪ { x } and I ∗ = I ∪ { y } and G ∗ can be constructedfrom G in polynomial time.Now let S be a DS of G with | S | ≤ k . Then S ∗ = S ∪ { x, y } is a SCDS of G ∗ with | S ∗ | ≤ k + 2 . Conversely, let S ∗ be a SCDS of G ∗ with | S ∗ | ≤ k + 2 . It follows from Proposition 2 that x, y ∈ S ∗ . Clearly S ′ = S ∗ \ { x, y } is a DS of G with | S ′ | ≤ k . Theorem 11.
STDM is NP-complete for split graphs.Proof.
It is clear that STDM for split graphs is in NP. The reduction given in the proof of Theorem10 shows that STDM is NP-complete for split graphs.Since the Domination problem is w[2]-complete for split graphs [8] and the reductions in Theorem 10and Theorem 11 are in the function of the parameter c, the following two corollaries are immediate. Corollary 3.
SCDM is w[2]-hard in split graphs.
Corollary 4.
STDM is w[2]-hard in split graphs.
5n the next two theorems we prove that γ sc ( G ) can be computed in linear time for block graphsand threshold graphs.Let G = ( V, E ) be a connected graph. A vertex v is called a cut-vertex of G if G − v is adisconnected graph. A graph G with no cut-vertex is called a block. A block B of a graph is amaximal connected induced subgraph of G such that B has no cut-vertex. In block B , verticeswhich are not cut vertices of G are called block vertices. A graph G is called a block graph if all itsblocks are cliques. Definition 1.
A graph G is called a block graph if all the blocks of G are cliques. Theorem 12.
Let G be a block graph having r blocks and k cut vertices. Then γ sc ( G ) = k + r − r ′ ,where r ′ is the number of blocks such that all vertices of the block are cut vertices.Proof. Let A denote the set of all cut vertices of G . Let B , B , . . . , B r ′ , B r ′ +1 , . . . , B r be the blocksof G where every vertex of B i is a cut vertex of G if 1 ≤ i ≤ r ′ . Let T = { v i : 1 ≤ i ≤ r − r ′ and v i is a non-cut vertex of B r ′ + i } . Let S = A ∪ T. Since A contains all cut-vertices of G , it followsthat G [ S ] is connected. Also if v ∈ V \ S , then v is not a cut-vertex. Now there exists a vertex u ∈ T such that uv ∈ E and ( S \ { v } ) ∪ { u } is a CDS of G. Hence, S is a SCDS of G. Therefore, γ sc ( G ) ≤ k + r − r ′ . Now let D be any γ sc -set of G. Since G [ D ] is connected, D ⊇ A. Further, a cut-vertex cannotdefend any other vertex and hence D contains at least one non-cut vertex from each block B i where r ′ + 1 ≤ i ≤ r. Hence γ sc ( G ) = | D | ≥ | S | = k + r − r ′ . Thus γ sc ( G ) = k + r − r ′ . Corollary 5.
Let G be a block graph with r blocks and exactly one cut-vertex. Then γ sc ( G ) = r + 1 . Corollary 6.
For any tree T with n vertices, γ sc ( T ) = n. Proof.
Here r = n − , r ′ = n − − l and k = n − l where l is the number of leaves in T. Therefore, γ sc ( T ) = n. Corollary 7.
SCDM is linear time solvable for block graphs.Proof.
Since number of blocks and number of cut-vertices of block graph can be determined inlinear time, the result follows.
Definition 2.
A graph G = ( V, E ) is called a threshold graph if there is a real number t and a realnumber w ( v ) for every v ∈ V such that a set S ⊆ V is independent if and only if P v ∈ S w ( S ) ≤ t . Threshold graphs considered are assumed to be non-complete and connected. We use the fol-lowing characterization of threshold graphs given in [6] to prove that secure connected dominationnumber can be computed in linear time for threshold graphs.A graph G is a threshold graph if and only if it is a split graph and for split partition ( C, I ) of V , there is an ordering ( x , x , . . . , x p ) of vertices of C such that N [ x ] ⊆ N [ x ] ⊆ . . . ⊆ N [ x p ] , andthere is an ordering ( y , y , . . . , y q ) of the vertices of I such that N ( y ) ⊇ N ( y ) ⊇ . . . N ( y q ) . Theorem 13.
Let G be a connected threshold graph. Then γ sc ( G ) = 2 + l , where l is the numberof pendant vertices. roof. Let S = { x p , x p − } ∪ { v ∈ I : v ∈ N ( x p ) \ N ( x p − ) } . Clearly G [ S ] is a star with center x p . Also every vertex v ∈ V \ S is defended by x p − and G [( S \ { x p − } ) ∪ { v } ] is connected. Thus, S is a SCDS of G. Hence γ sc ( G ) ≤ l. Let D be any γ sc -set of G . It follows from Theorem 2 that | D | ≥ l + 1 . If | D | = l + 1 , thenexactly one vertex of C say, u is a support vertex. Hence no vertex of C \ { u } is D -defended, whichis a contradiction. Hence γ sc ( G ) = | D | ≥ l. Thus γ sc ( G ) = 2 + l. Theorem 14.
SCDM is linear time solvable for threshold graphs.Proof.
Since the ordering of the vertices of the clique in a threshold graph can be determined inlinear time [6], the result follows.
3. Conclusion
In this paper, it is shown that secure connected (total) domination problem is NP-complete evenwhen restricted to bipartite graphs, or split graphs. Since split graphs form a proper subclass ofchordal graphs, these problems are also NP-complete for chordal graphs. We have proved that secureconnected domination problem is linear time solvable for block graphs and threshold graphs. Itwill be interesting to investigate the algorithmic complexity of secure connected (total) dominationproblem for subclasses of chordal and bipartite graphs.
References [1] A.A. Bertossi,
Dominating sets for split and bipartite graphs.
Information Processing Letters, (1984), 37–40.[2] A.G. Cababro, S.S. Canoy Jr., and I.S. Aniversario,
Secure Connected Domination in a Graph ,International Journal of Mathematical Analysis (2014), 2065-2074.[3] A.G. Cabaro, and S.R. Canoy Jr., Secure Connected Dominating Sets in the Join and Compo-sition of Graphs , International Journal of Mathematical Analysis (2015), 1241-1248.[4] E.J. Cockayne, P.J.P. Grobler, W.R. Grundlingh, J. Munganga, and J.H. van Vuuren,
Protec-tion of a Graph , Utilitas Mathematica (2005), 19-32.[5] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness , Freeman, New York, (1979).[6] N. Mahadev, U. Peled,
Threshold Graphs and Related Topics, in: Annals of Discrete Mathe-matics , vol. 56, North Holland, (1995).[7] J. Pfaff, R. Laskar, and S.T. Hedetniemi,
NP-completeness of Total and Connected Dominationand Irredundance for Bipartite Graphs , Technical Report , Clemson University Clemson,SC, (1983).[8] V. Raman, and S. Saurabh.
Short cycles make W-hard problems hard: FPT algorithms forW-hard problems in graphs with no short cycles , Algorithmica (2008), 203-225.[9] D.B. West,