Semipaired Domination in Some Subclasses of Chordal Graphs
SSemipaired Domination in Some Subclasses of Chordal Graphs
Michael A. Henning ∗ , Arti Pandey † , and Vikash Tripathi ‡ Department of Mathematics and Applied Mathematics, University of Johannesburg AucklandPark, 2006 South Africa Department of Mathematics, Indian Institute of Technology Ropar, Nangal Road, Rupnagar,Punjab 140001, INDIA
Abstract
A dominating set D of a graph G without isolated vertices is called semipaired dominating set if D can be partitioned into -element subsets such that the vertices in each set are at distance at most .The semipaired domination number, denoted by γ pr ( G ) is the minimum cardinality of a semipaireddominating set of G . Given a graph G with no isolated vertices, the M INIMUM S EMIPAIRED D OM - INATION problem is to find a semipaired dominating set of G of cardinality γ pr ( G ) . The decisionversion of the M INIMUM S EMIPAIRED D OMINATION problem is already known to be NP-completefor chordal graphs, an important graph class. In this paper, we show that the decision version of theM
INIMUM S EMIPAIRED D OMINATION problem remains NP-complete for split graphs, a subclassof chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimumcardinality semipaired dominating set of block graphs. In addition, we prove that the M
INIMUM S EMIPAIRED D OMINATION problem is APX-complete for graphs with maximum degree . Keywords: Domination, Semipaired domination, Block graphs, NP-completeness, Graph algorithms.
For a graph G = ( V, E ) , a vertex v ∈ V is said to dominate a vertex w ∈ V if either v = w or vw ∈ E . A dominating set of G is a set D ⊆ V such that every vertex in V is dominated byat least one vertex of D . The minimum cardinality of a dominating set of G is called the dominationnumber of G denoted by γ ( G ) . Many facility location problems can be modelled using the concept ofdomination in graphs. Due to vast applications of domination, many variations of dominations have alsobeen introduced in the literature. The domination problem are widely studied from combinatorial as wellas algorithmic point of view, see [6, 7].One important variation of domination is paired domination. The concept of paired domination wasintroduced by Haynes and Slater in [5]. For a graph G with no isolated vertices, a dominating set D iscalled a paired dominating set , abbreviated as PD-set, if the graph induced by D has a perfect matching M . Two vertices joined by an edge of M are said to be paired and are also called partners in D . ∗ [email protected] † [email protected] ‡ a r X i v : . [ c s . D M ] A ug he M INIMUM PAIRED DOMINATION problem is to find a PD-set of G of minimum cardinality. Thecardinality of such a set is known as the paired domination number of G , and is denoted by γ pr ( G ) . Asurvey of paired domination can be found in [13].A relaxed version of paired domination, known as semipaired domination was introduced by Haynesand Henning [3], and further studied by others [4, 8, 9, 10, 14]. For a graph G with no isolated vertex, a semipaired dominating set , abbreviated as semi-PD-set, is a dominating set D of G such that the verticesin D can be partitioned into -sets such that if { u, v } is a -set, then uv ∈ E ( G ) or the distance between u and v is . We say that u and v are semipaired , and that u and v are partners . The minimum cardinalityof a semi-PD-set of G is called the semipaired domination number of G , and is denoted by γ pr ( G ) .For a graph G with no isolated vertices, the M INIMUM S EMIPAIRED D OMINATION problem is to finda semi-PD-set of cardinality γ pr ( G ) . For a given graph G and a positive integer k , the S EMIPAIRED D OMINATION D ECISION problem is to determine whether G has a semi-PD-set of cardinality at most k or not. Since every PD-set is a semi-PD-set, and since every semi-PD-set is a dominating set, we havethe following observation. Observation 1.1. ([3]) For every graph G without isolated vertices, γ ( G ) ≤ γ pr ( G ) ≤ γ pr ( G ) . The algorithmic study of the M
INIMUM S EMIPAIRED D OMINATION problem was initiated by Hen-ning et al. in [10]. They proved that the S
EMIPAIRED D OMINATION D ECISION problem is NP-completeeven for bipartite graphs and chordal graphs. They also proposed a linear-time algorithm to compute aminimum cardinality semi-PD-set of an interval graph. They proposed a -approximationalgorithm for the M
INIMUM S EMIPAIRED D OMINATION problem, where ∆ denotes the maximum de-gree of the graph. On the negative side, they proved that M INIMUM S EMIPAIRED D OMINATION problemcannot be approximated within (1 − (cid:15) ) ln | V | for any (cid:15) > unless P = NP. In this paper, we continue thealgorithmic study of the M
INIMUM S EMIPAIRED D OMINATION problem. The main contributions of thepaper are summarized below.In Section 2, we discuss some definitions and notations. In this section we also observe some graphclasses where paired domination and semipaired domination problems differ in complexity. In Section , we prove the S EMIPAIRED D OMINATION D ECISION problem is NP-complete for split graphs. InSection , we propose a linear time algorithm to compute a minimum cardinality semipaired dominatingset in block graphs. In Section 5, we show that the M INIMUM S EMIPAIRED D OMINATION problem isAPX-hard for graphs with maximum degree . Finally, Section concludes the paper. Let G = ( V, E ) be a graph. For a vertex v ∈ V , let N G ( v ) = { u ∈ V | uv ∈ E } and N G [ v ] = N G ( v ) ∪ { v } denote the open neighborhood and the closed neighborhood of v , respectively. For a set S ⊆ V , the sets N G ( S ) = (cid:83) u ∈ S N G ( u ) and N G [ S ] = N G ( S ) ∪ S are called open neighborhood andthe closed neighborhood of S , respectively. For a set S ⊂ V , the graph G \ S is obtained from G bydeleting all vertices in S and all edges incident with vertices in S . If S = { v } , we write G \ v ratherthan G \ { v } . A cut vertex in a connected graph G is a vertex v ∈ V such that G \ v is disconnected.Let n and m denote the number of vertices and edges of G , respectively. We use standard notation, [ k ] = { , , . . . , k } . In this paper, we only consider connected graphs with at least two vertices.2 set S ⊆ V is called an independent set of G if uv / ∈ E for all u, v ∈ S . A set K ⊆ V is calleda clique of G if uv ∈ E for all u, v ∈ K . A graph G is said to be a chordal graph if every cycle in G of length at least four has a chord , that is, an edge joining two non-consecutive vertices of the cycle. Achordal graph G is a split graph if V can be partitioned into two sets C and I such that C is a clique and I is an independent set.A rooted tree T distinguishes one vertex r called the root . For each vertex v (cid:54) = r of T , the parent of v is the neighbor of v on the unique ( r, v ) -path, while a child of v is any other neighbor of v . Further,the grandparent of v is the vertex at distance from v on the unique ( r, v ) -path. A descendant of v is avertex u (cid:54) = v such that the unique ( r, u ) -path contains v . A grandchild of v in T is a descendant of v atdistance from v . In this subsection, we discuss the complexity difference between paired domination and semipaireddomination. We show that for the class of GP graphs, which we define below, the decision version ofthe M INIMUM P AIRED D OMINATION problem is NP-complete, but the M
INIMUM S EMIPAIRED D OM - INATION problem is easily solvable. On the other hand, we introduce a graph class called GP graphs,and we show that the S EMIPAIRED D OMINATION D ECISION problem is NP-complete for GP graphs,but the M INIMUM P AIRED D OMINATION problem is easily solvable for this graph class.The class of GP graphs was introduced by Henning et al. in [11]. Below we recall the definition ofGP graphs. Definition 2.1 (GP -graph) . A graph G = ( V, E ) is called a GP - graph if it can be obtained from ageneral connected graph H = ( V H , E H ) where V H = { v , v , . . . , v n H } , by adding a path of length toevery vertex of H . Formally, V = V H ∪{ w i , x i , y i , z i | i ∈ [ n H ] } and E = E H ∪{ v i w i , w i x i , x i y i , y i z i | i ∈ [ n H ] } , where n H denotes the number of vertices in H . Theorem 2.1. If G is a GP - graph , then γ pr ( G ) = | V ( G ) | .Proof. Let G be a GP -graph of order n = | V ( G ) | = 5 | V H | as constructed in Definition 2.1. Theset S = { w i , y i | i ∈ [ n H ] } is a semi-PD-set of G , implying that γ pr ( G ) ≤ | V ( G ) | . Every semi-PD-set of G must contain at least two vertices from the set { w i , x i , y i , z i } for each i ∈ [ n H ] . Thus, γ pr ( G ) ≥ | V ( G ) | . Consequently, γ pr ( G ) = | V ( G ) | . Lemma 2.1. If G is a GP - graph constructed from a graph H as in Definition 2.1, then H has a PD-setof cardinality at most k , k ≤ n H , if and only if G has a PD-set of cardinality at most n H + k .Proof. Suppose D is a PD-set of H of cardinality at most k . Then the set D ∪ { x i , y i | i ∈ [ n H ] } is aPD-set of G of cardinality at most n H + k . Conversely, assume that G has a PD-set D (cid:48) of cardinalityat most n H + k . In order to dominate z i , the set D (cid:48) must contain either y i or z i . Without loss ofgenerality, we may assume that D (cid:48) contains y i and that x i ∈ D (cid:48) with x i and y i paired in D (cid:48) . Thisimplies that D (cid:48) contains exactly two vertices from the set { x i , y i , z i } , where i ∈ [ n H ] . Now we considerthe set D = D (cid:48) \ { x i , y i , z i | i ∈ [ n H ] } . We note that | D | ≤ k and D ⊆ { v i , w i | i ∈ [ n H ] } .Further, the set D dominates V H , and for every vertex u in D , the partner of u is also present in D . Let G (cid:48) = G [ V ( H ) ∪ { w i | i ∈ [ n H ] } ] . Observe that w i can be paired only with v i in G (cid:48) for i ∈ [ n H ] . If w i ∈ D and N ( v i ) ⊆ D , then we update the set D as D \ { w i , v i } . If there exist a vertex u ∈ N ( v i ) suchthat u / ∈ D , then we update the set D as D = ( D \ { w i } ) ∪ { u } . We do this update for each w i ∈ D .Note that the updated set D is a PD-set of H and | D | ≤ k . Hence, the result follows.3ince the decision version of the M INIMUM P AIRED D OMINATION problem is known to be NP-complete for general graphs [5], the following theorem follows directly from Lemma 2.1.
Theorem 2.2.
The decision version of the M INIMUM P AIRED D OMINATION problem is NP-completefor GP - graphs . Next, we define a new graph class and call it GP -graphs. Definition 2.2 (GP -graph) . A graph G = ( V, E ) is called a GP - graph if it can be obtained from ageneral connected graph H = ( V H , E H ) where V H = { v , v , . . . , v n } , by adding a vertex disjointpath P for each vertex v of H and joining v to the central vertex of the path. Formally, V = V H ∪{ a i , b i , c i , d i , e i | i ∈ [ n ] } and E = E H ∪ { v i c i , c i b i , c i d i , b i a i , d i e i | i ∈ [ n ] } , where n denotes thenumber of vertices in H .For example, when H is a -cycle C , then a GP graph obtained from H is shown in Fig. 1. For aGP graph, we show that the S EMIPAIRED D OMINATION D ECISION problem is NP-complete, but theM
INIMUM P AIRED D OMINATION problem is easily solvable. b bb b b bb bb bbb b b b bb bb bbb bb bbbb v v v v v v v v c c c c b b b b a a a a d d d d e e e e H GP Figure 1: An illustration of a GP graph obtained from a -cycle. Theorem 2.3. If G is a GP - graph , then γ pr ( G ) = | V ( G ) | .Proof. Let G be a GP -graph of order n = | V ( G ) | = 6 | V H | as constructed in Definition 2.2. The set S = { a i , b i , c i , d i | i ∈ [ n H ] } is a PD-set of G , implying that γ pr ( G ) ≤ | V ( G ) | . Also we note thatevery PD-set of G must contain at least four vertices from the set { a i , b i , c i , d i , e i } for each i ∈ [ n H ] .Hence, γ pr ( G ) ≥ | V ( G ) | . Consequently, γ pr ( G ) = | V ( G ) | . Lemma 2.2. If G is a GP - graph constructed from a graph H as in Definition 2.2, then H has a semi-PD-set of cardinality k , k ≤ n H if and only if G has a semi-PD-set of cardinality n H + k .Proof. Suppose D is semi-PD-set of H of cardinality at most k . Then the set D ∪ { b i , d i | i ∈ [ n H ] } is a semi-PD-set of G of cardinality at most n H + k . Conversely, assume that G has a semi-PD-set D (cid:48) of cardinality at most n H + k . In order to dominate a i , the set D (cid:48) must contain either a i or b i .Similarly, in order to dominate e i , the set D (cid:48) must contain either a i or b i . Without loss of generality, foreach i ∈ [ n H ] we may assume that D (cid:48) contains { b i , d i } with b i and d i semipaired in D (cid:48) . This implies4hat D (cid:48) contains exactly two vertices from the set { a i , b i , d i , e i } , where i ∈ [ n H ] . Now we considerthe set D = D (cid:48) \ { a i , b i , d i , e i | i ∈ [ n H ] } . We note that | D | ≤ k and D ⊆ { v i , c i | i ∈ [ n H ] } .Further, the set D dominates V H , and for every vertex u in D , the semipair of u is also present in D . Let G (cid:48) = G [ V ( H ) ∪ { c i | i ∈ [ n H ] } ] . If c i ∈ D and c i is semipaired with a vertex v i such that N ( v i ) ⊆ D ,then we update the set D as D \ { c i , v i } and if there exist a vertex u ∈ N ( v i ) such that u / ∈ D , then weupdate the set D as D = ( D \ { c i } ) ∪ { u } . We do this update for each c i ∈ D . Note that the updated set D is a semi-PD-set of H and | D | ≤ k . Hence, the result follows.Since the S EMIPAIRED D OMINATION D ECISION problem is known to be NP-complete for generalgraphs [10], the following theorem follows directly from Lemma 2.2.
Theorem 2.4.
The S EMIAIRED D OMINATION D ECISION problem is NP-complete for GP - graphs . In this section, we prove that the S
EMIPAIRED D OMINATION D ECISION problem is NP-completefor split graphs. To prove this NP-completeness result, we use a reduction from the domination problem,which is a well known NP-complete problem [6].
Theorem 3.1.
The S EMIPAIRED D OMINATION D ECISION problem is NP-complete for split graphs.Proof.
Clearly, the S
EMIPAIRED D OMINATION D ECISION problem is in NP. To show the hardness, wegive a polynomial time reduction from the D
OMINATION D ECISION problem for general graphs. Givena non-trivial graph G = ( V, E ) , where V = { v i | i ∈ [ n ] } and E = { e j | j ∈ [ m ] } , we construct a splitgraph G (cid:48) = ( V G (cid:48) , E G (cid:48) ) as follows:Let V k = { v ki | i ∈ [ n ] } and U k = { u ki | i ∈ [ n ] } for k ∈ [2] . Now define V G (cid:48) = V ∪ V ∪ U ∪ U ,and E G (cid:48) = { uv | u, v ∈ V ∪ U , u (cid:54) = v } ∪ { v i v j , u i u j | i ∈ [ n ] and v j ∈ N G [ v i ] } . Note that the set A = V ∪ U is a clique in G (cid:48) and the set B = V ∪ U is an independent set in G (cid:48) . Since V G (cid:48) = A ∪ B ,the constructed graph G (cid:48) is a split graph. Fig. 2 illustrates the construction of G (cid:48) from G . b bbb b bb bbb bbbbbbbbbb v v v v v v v v u u u u u u u u v v v v G G ′ Figure 2: An illustration to the construction of G (cid:48) from G in the proof of Theorem 3.1.Now, to complete the proof of the theorem, we only need to prove the following claim. Claim 3.1.
The graph G has a dominating set of cardinality at most k if and only if G (cid:48) has a semi-PD-setof cardinality at most k . roof. Let D = { v i , v i , . . . , v i k } be a dominating set of G of cardinality at most k . Then the set D (cid:48) = { v i , v i , . . . , v i k } ∪ { u i , u i , . . . , u i k } is a semi-PD-set of G (cid:48) of cardinality at most k .Conversely, let D (cid:48) is a semi-PD-set of G (cid:48) of cardinality at most k . Then, either | D (cid:48) ∩ ( V ∪ V ) | ≤ k or | D (cid:48) ∩ ( U ∪ U ) | ≤ k . Without loss of generality, we assume that | D (cid:48) ∩ ( V ∪ V ) | ≤ k . Let D = D (cid:48) ∩ ( V ∪ V ) . Note that V is an independent set, hence any vertex v ∈ V is either dominated by itselfor by some vertex in V . If v i ∈ D and none of its neighbors is in D , then update D = ( D \ { v i } ) ∪ { u } where u ∈ N ( v i ) . We do this update for each vertex v i ∈ V . Now observe that in the updated set D ,we have N ( v i ) ∩ D (cid:54) = ∅ for i ∈ [ n ] . The set D (cid:48)(cid:48) = { v i | v i ∈ D } is a dominating set of G of cardinalityat most k . This proves the claim.Hence the result follows. For a graph G , a maximal induced subgraph of G without a cut vertex is called a block of G . If B and B (cid:48) are two blocks of G then | V ( B ) ∩ V ( B (cid:48) ) | ≤ , and a vertex v ∈ V ( B ) ∩ V ( B (cid:48) ) if and only if v is a cut vertex. A connected graph whose every block is a complete graph is called a block graph . A treeis block graph in which every block contains exactly two vertices. A block with only one cut vertex iscalled an end block . Every block graph not isomorphic to a complete graph has at least two end blocks.Lie Chen et al. [12] have studied an ordering of vertices of block graph, α = ( v , v , . . . , v n ) suchthat v i v j ∈ E and v i v k ∈ E implies v j v k ∈ E for i < j < k ≤ n . Such an ordering of vertices ofa block graph is called Block-Elimination-Ordering ( BEO ) . The procedure to get such an ordering is asfollows: if G is not isomorphic to a complete graph, then it must have at least two end blocks. Pick anend block, say B , having a cut vertex x . Staring with the index , enumerate the vertices in V ( B ) \ { x } in any order and remove V ( B ) \ { x } from the graph. Let k = max { s | v s ∈ V ( B ) \ { x }} , that is, v k is the vertex in V ( B ) \ { x } having highest index. Now if the remaining graph, say G (cid:48) , is a completegraph, then enumerate the remaining vertices starting from index k + 1 to n in any order; otherwise, pickan end block in G (cid:48) , say B (cid:48) , having cut vertex x (cid:48) . Starting with index k + 1 enumerate the vertices in V ( B (cid:48) ) \ { x (cid:48) } and continue the procedure in G (cid:48) \ ( V ( B (cid:48) ) \ { x (cid:48) } ) .Let G = ( V, E ) be a block graph, and α = ( v , v , . . . , v n ) be a BEO of vertices of G . For i (cid:54) = n , wedefine F ( v i ) = v j , where j = max { k | v i v k ∈ E } . We also define F ( v n ) = n . Further, we constructa block tree T ( G ) rooted at v n such that V ( T ( G )) = V ( G ) and E ( T ( G )) = { uv if and only if either F ( u ) = v or F ( v ) = u } . Fig. 3 illustrates the construction of T ( G ) from a block graph G . Note that acut vertex of G is an internal vertex of T ( G ) . Also if B is a block of G with V ( B ) = { u i , u i , . . . , u i k } where u i k is the highest index vertex in V ( B ) , then u i , u i , . . . , u i k − are called siblings in T ( G ) andeach one is a child of u i k . The following complexity is already known. Theorem 4.1. [12] For a block graph G = ( V, E ) , a BEO can be computed in O ( n + m ) -time. Inaddition, given a BEO, the corresponding block tree can also be computed in O ( n + m ) -time. Observation 4.1.
Let G = ( V ( G ) , E ( G )) be a block graph and T ( G ) be a corresponding block tree. If v ∈ N G ( u ) , then one of the following holds in T ( G ) , (a) u is a parent of v . (b) u is a child of v . bb bbbbb b b b bb b bbb b u u u u u u u u u u u u u u u u u u u bb b b b b bb bbbb bbb b bb b b u u u u u u u u u u u u u u u u u u u b u b u G T ( G ) Figure 3: An illustration of the construction of T ( G ) from a block graph G .(c) u is a sibling of v . Lemma 4.1.
Let G be a block graph with given BEO, α = ( v , v , . . . , v n ) . If B and B (cid:48) are anytwo blocks of G such that v i ∈ V ( B ) ∩ V ( B (cid:48) ) , then F ( u ) = v i , for all u ∈ V ( B ) \ { v i } or for all u ∈ V ( B (cid:48) ) \ { v i } .Proof. Let G be a block graph with given BEO, α = ( v , v , . . . , v n ) . Suppose B and B (cid:48) are blocks of G such that v i ∈ V ( B ) ∩ V ( B (cid:48) ) . Clearly, v i is a cut vertex. By the way the vertices of G are enumerated,either all the vertices in V ( B ) \ { v i } first get enumerated and thereafter the vertices in V ( B (cid:48) ) , or allthe vertices in V ( B (cid:48) ) \ { v i } first get enumerated and then the vertices in V ( B ) . Renaming the blocksif necessary, we may assume without loss of generality that the vertices in V ( B ) \ { v i } get enumeratedfirst and thereafter the vertices in V ( B (cid:48) ) . In that case, we note that i ≥ max { r | v r ∈ V ( B ) \ { v i }} .Thus, F ( u ) = v i for all u ∈ V ( B ) \ { v i } , implying the desired result.Using above lemma, we state that if there are exactly k blocks, say B , B . . . B k of a block graph G such that v i ∈ V ( B ) ∩ V ( B ) ∩ · · · V ( ∩ B k ) . Let B ∗ ∈ { B , B . . . B k } be the block whose verticesare enumerated after the enumeration of vertices in { V ( B ) ∪ V ( B ) ∪ · · · ∪ V ( B k ) } \ { V ( B ∗ ) } in theBEO α . Then, for every vertex v j ∈ { V ( B ) ∪ V ( B ) ∪ · · · ∪ V ( B k ) } \ { V ( B ∗ ) } , F ( v j ) = v i . Lemma 4.2.
Let G be a block graph and T ( G ) be a block tree of G . If α = ( v n , v n − , . . . , v ) is aBFS-ordering of the vertices of T ( G ) rooted at v n then the reverse of BFS-ordering β = ( v , v , . . . , v n ) also satisfy BEO in G .Proof. Let G be a block graph and T ( G ) be the corresponding block tree. Let β = ( v , v , . . . , v n ) be the reverse of BFS-ordering α = ( v n , v n − , . . . , v ) of vertices of G as they appear in T ( G ) . For i < j < k , let v i , v j and v k satisfy the reverse of BFS-ordering and v i v j , v i v k ∈ E ( G ) . To prove theresult we need to show v j v k ∈ E ( G ) . By contradiction, suppose v j v k / ∈ E ( G ) . Since v j v k / ∈ E ( G ) ,this implies v j and v k belongs to different blocks of G , say B and B (cid:48) respectively, and v i is a cut vertex.Now, using Lemma 4.1 we have F ( u ) = v i , for all u ∈ V ( B ) \ { v i } or for all u ∈ V ( B (cid:48) ) \ { v i } .Therefore in T ( G ) , the vertex v i is the parent of either v j or v k , which is a contradiction as v i , v j and v k satisfy the reverse of BFS-ordering and i < j < k . Hence the result follows.7et G = ( V ( G ) , E ( G )) be a block graph and T ( G ) be a corresponding block tree of G . Let α =( v n , v n − , . . . , v ) be a BF S -ordering of vertices of T ( G ) rooted at v n . Recall that the distance betweentwo vertices u and v is the length of shortest path between u and v , denote by d ( u, v ) . For a positiveinteger l ≥ , We say a vertex x is at level l in a tree T rooted at a vertex y , if d T ( x, y ) = l . In ouralgorithm, we will process the vertices of the block graph as they appear in the reverse of BF S -ordering β = ( v , v , . . . , v n ) of the corresponding block tree T ( G ) . We will use the following notation whileprocessing the vertices in the algorithm: D ( v i ) = (cid:40) if v i is not dominated , if v i is dominated .L ( v i ) = if v i is not selected , if v i is selected but not semipaired , if v i is selected and semipaired .m ( v i ) = (cid:40) k if v k needs to semipaired with a vertex in N T ( G ) [ v i ] or with some sibling of v i , otherwise . Also we use N i ( v k ) = { v j | v k v j ∈ E ( G ) and j ≥ i } and N i [ v k ] = { v j | v k v j ∈ E ( G ) and j ≥ i } ∪ { v k } . Lemma 4.3.
Let G be a block graph, and T ( G ) be a block tree of G . Let β = ( v , v , . . . , v n ) be the reverse of BF S -ordering of vertices in T ( G ) , then v i v j ∈ E ( G ) implies N i [ v i ] ⊆ N i [ v j ] where j ≥ i .Proof. If v j is parent of v i in T ( G ) , then a neighbor v k of v i with k > i is a sibling of v i in T ( G ) . Hence, v k v j ∈ E ( G ) and the result follows. If v j is a sibling of v i and v i v j ∈ E ( G ) , then v i and v j are in the same block B of G . Now any neighbor v k of v i with k > i is either a sibling or parent of v i in T ( G ) . If v k is parent of v i then v j v k ∈ E ( G ) (as v j is a sibling of v i ). Next we consider that v k is a sibling of v i . Note that either i < j < k or i < k < j , and the vertices v i , v j , v k appears in the reverse of BFS-ordering, which is also a BEO (by Lemma 4.2).Since v i v k , v i v j ∈ E ( G ) , v j v k ∈ E ( G ) by using the property of BEO. Observation 4.2.
If a vertex v i is at level l + 2 , then v i does not have any neighbor at level l . Also if v i issemipaired with v j , then v j may be one of the following in T ( G ) . ( i ) Parent of v i that is F ( v i ) or a grand parent of v i that is F ( F ( v i )) . ( ii ) Child of v i or a grand child of v i . ( iii ) Sibling of v i or sibling of F ( v i ) . ( iv ) Child of some sibling v s of v i . Next, we present the detailed algorithm to compute a minimum cardinality semi-PD-set of a given block graph. lgorithm 1 Minimum Semipaired Domination in Block GraphsInput: A block graph G=(V,E), corresponding block-tree T ( G ) , reverse of BFS ordering of vertices of T ( G ) : β = ( v , v , . . . , v n ) Output:
A minimum cardinality semi-PD-set D sp of G . for i = 1 to n doif ( D ( v i ) = 0 and i (cid:54) = n ) then D sp = D sp ∪ { F ( v i ) } , let F ( v i ) = v j ; L ( v j ) = 1 ; D ( u ) = 1 ∀ u ∈ N G [ v j ] ;Let C = { u ∈ N G [ v j ] | m ( u ) (cid:54) = 0 } ; if ( C = ∅ ) then m ( F ( v j )) = j ; else Let k = min { b | v b ∈ C } and let m ( v k ) = r ; L ( v j ) = L ( v r ) = 2 ; // semipair v j with v r m ( v k ) = 0 ; if ( D ( v i ) (cid:54) = 0 and m ( v i ) = k (cid:54) = 0) thenif ( L ( F ( v i )) = 0) then D sp = D sp ∪ { F ( v i ) } ; D ( u ) = 1 ∀ u ∈ N G [ F ( v i )] ; L ( v k ) = L ( F ( v i )) = 2 and m ( v i ) = 0 ; // semipair v k with F ( v i ) ; else if ( L ( v i ) = 0) then D sp = D sp ∪ { v i } ; L ( v k ) = L ( v i ) = 2 ; // semipair v k with v i D ( u ) = 1 ∀ u ∈ N G [ v i ] ; m ( v i ) = 0 ; else D sp = D sp ∪ { u } where u ∈ N ( v k ) and L ( u ) = 0 ; L ( v k ) = L ( u ) = 2 and m ( v i ) = 0 ; // semipair v k with u ; if ( D ( v n ) = 0) then L ( v n ) = 2 ; D ( v n ) = 1 ; L ( u ) = 2 for some u ∈ N G ( v n ) with L ( u ) = 0 ; // semipair v n with uD sp = D sp ∪ { v n , u } ;return D sp ; Illustration of the Algorithm with an example
We illustrate the algorithm for computing a minimum cardinality semi-PD-set of the block graph shown inFig. 3. Since there are vertices in the graph, the algorithm will terminate in -iterations. We process thevertices as they appear in the reverse of BFS-ordering. For the graph in the Fig. 3, a reverse of BFS-ordering ofthe vertices is given by β = ( u , u , u , u , u , u , u , u , u , u , u , u , u , u , u , u , u , u , u , u ) .The iterations of the algorithm are as follows: NITIALLY D ( u i ) = L ( u i ) = m ( u i ) = 0 for u ∈ [20] and D sp = ∅ .I TERATION Since D ( u ) = 0 , therefore we select F ( u ) = u . L ( u ) = 1 and D ( u ) = D ( u ) = D ( u ) = D ( u ) = D ( u ) = D ( u ) = 1 As C = ∅ , we have m ( F ( u )) = m ( u ) = 7 A FTER I TERATION : D sp = { u } I TERATION Since D ( u ) = 0 , therefore we select F ( u ) = u . L ( u ) = 1 and D ( u ) = D ( u ) = 1 In this iteration note that C = { u } , k = 12 and m ( u ) = 10 . L ( u ) = L ( u ) = 2 and m ( u ) = 0 A FTER I TERATION : D sp = { u , u } We do not have any update in I
TERATION , , , , AND I TERATION Since D ( u ) = 0 , therefore we select F ( u ) = u . L ( u ) = 1 and D ( u ) = D ( u ) = D ( u ) = D ( u ) = 1 As C = ∅ , hence m ( F ( u )) = m ( u ) = 15 A FTER I TERATION : D sp = { u , u , u } We do not have any update in I
TERATION I TERATION Since D ( u ) = 0 , therefore we select F ( u ) = u . L ( u ) = 1 and D ( u ) = D ( u ) = D ( u ) = D ( u ) = D ( u ) = 1 As C = ∅ , hence m ( F ( u )) = m ( u ) = 19 A FTER I TERATION : D sp = { u , u , u , u } We do not have any update in I
TERATION AND I TERATION Since D ( u ) = 0 , therefore we select F ( u ) = u . L ( u ) = 1 and D ( u ) = D ( u ) = D ( u ) = D ( u ) = 1 As C = { u } , k = 20 and m ( u ) = 19 hence L ( u ) = L ( u ) = 2 and m ( u ) = 0 A FTER I TERATION : D sp = { u , u , u , u , u } We do not have any update in I
TERATION , , AND I TERATION In this iteration note that m ( u ) = 15 (cid:54) = 0 .As L ( F ( u )) = L ( u ) = 0 hence, L ( u ) = L ( u ) = 2 and m ( u ) = 0 .A FTER I TERATION : D sp = { u , u , u , u , u , u } We do not have any update in I
TERATION AND Lemma 4.4.
For ≤ i ≤ n , when v i is the currently considered vertex, the following holds:1. D ( v j ) = 1 for j ∈ [ i − .2. m ( v j ) = 0 for j ∈ [ i − .3. if v i is at level l from the root in T ( G ) , then for all vertices u at level l + 2 or above, either L ( u ) = 0 or .Proof. The proof directly follows from Algorithm . Observation 4.3.
If in the i th -iteration, m ( v i ) = k (cid:54) = 0 , then for any u ∈ N G [ v i ] \ { v k } , L ( u ) = 0 or . For each ≤ i ≤ n , let D i = { v | L ( v ) > } when v i has just been considered. In particular D = ∅ and D n = { v | L ( v ) > } , when all the vertices of the graph have been processed. Clearly, D n is a dominating set.Note that when we are processing the vertex v n in our algorithm, by Lemma 4.4 for all vertices u at level or bove, either L ( u ) = 0 or and D ( v j ) = 1 for j ∈ [ n − . Also note that in the n th -iteration, if for a vertex u atlevel , we have L ( u ) = 1 , then m ( v n ) (cid:54) = 0 and hence by Observation 4.3 for all vertices w other than u at level , we have L ( w ) = 0 or . In this iteration, if L ( v n ) = 0 , then we semipair u with v n ; otherwise, we semipair u with one of its neighbours. Also, if D ( v n ) = 0 , this implies no vertex from level is selected in the dominatingset. So, according to Algorithm , we select v n in the semi-PD-set and pair it with one of it’s children. Therefore,after the n th -iteration for all u ∈ V ( G ) , we have L ( u ) = 0 or and D n is a semi-PD-set. So in order to prove thecorrectness of the algorithm, we need to show that D n is contained in a minimum semi-PD-set D ∗ sp . Lemma 4.5.
For each ≤ i ≤ n , there is minimum semi-PD-set D (cid:48) sp such that:1. D i ⊆ D (cid:48) sp and if u and v are semipaired in D i then u and v are also semipaired in D (cid:48) sp .2. if in the i th -iteration we are updating m ( v j ) = k for some vertex v j , then v k is either semipaired with avertex in N T ( G ) [ v j ] or with a sibling of v j in D (cid:48) sp .Proof. We will prove the result using induction on i . Clearly, when i = 0 , there is semi-PD-set D (cid:48) sp such that D ⊆ D (cid:48) sp . Now, suppose that the result holds for any integer less than i < n , that is, there is a minimum semi-PD-set D ∗ sp such that D i − ⊆ D ∗ sp and the second condition of the lemma is satisfied.Now in the i th -iteration we will have the following cases: Case 1. D ( v i ) = 0 and i (cid:54) = n .In this case, D i = D i − ∪ { F ( v i ) } . Let F ( v i ) = v j and assume that v i is at level l ≥ from the root v n . Weproceed further with the following claim. Claim 4.1.
Let v j / ∈ D ∗ sp , v p ∈ D ∗ sp be a vertex dominating v i , and v q be the vertex which is semipaired with v p in D ∗ sp , then either ( D ∗ sp \ { v q } ) ∪ { v j } or ( D ∗ sp \ { v p } ) ∪ { v j } is a minimum semi-PD-set satisfying the firststatement of the lemma.Proof. Suppose that v j / ∈ D ∗ sp . Let v p be a vertex in D ∗ sp dominating v i , and let v q be the vertex that is semipairedwith v p . Clearly, j / ∈ { p, q } and v p / ∈ D i − . If v q ∈ D i − and v q is at level l + 2 or more, then v p ∈ D i − byLemma 4.4, a contradiction. Hence, if v q ∈ D i − , then v q is at level l + 1 or less from the root in T ( G ) . Further if v q / ∈ D i − and v q is at level l + 2 or more in T ( G ) , then by Lemma 4.4, the set D (cid:48) sp = ( D ∗ sp \ { v q } ) ∪ { v j } where v j is semipaired with v p is the required minimum semi-PD-set. Hence, we may assume that v q is at level l + 1 orless.Suppose firstly that the vertex v p is a child of v i in T ( G ) . Since v q is at level l + 1 or less and q (cid:54) = j , wehave v q ∈ N G [ v i ] . Hence, d ( v j , v q ) ≤ . Note that N i ( v p ) ⊆ N i ( v j ) . Since d ( v q , v j ) ≤ , the set D (cid:48) sp = D ∗ sp \ { v p } ∪ { v j } where v j is semipaired with v q is the required minimum semi-PD-set.Suppose secondly that v p is a sibling of v i or v p = v i in T ( G ) . In this case, v p is a child of v j in T ( G ) . Since v q is at level l + 1 or less hence, either v q is a child of v p or a child of some sibling say v s of v p or v q is a siblingof v p or a sibling of v j or v q = F ( v j ) .If v q is a sibling of v p or a sibling of v j or v q = F ( v j ) , we have v q v j ∈ E ( G ) . If v q is a child of v p , then d ( v j , v q ) = 2 . If v q is a child of some sibling, say v s , of v p , then v s is a child of v j . Thus since v q is a child of v s in T ( G ) , we again have d ( v j , v q ) = 2 . In all of the above cases, we have d ( v j , v q ) ≤ . Further, since v p is achild of v j , using Lemma 4.4 and the fact that N i [ v p ] ⊆ N i [ v j ] , we have that D (cid:48) sp = ( D ∗ sp \ { v p } ) ∪ { v j } where v j is semipaired with v q is the required minimum semi-PD-set.By Claim 4.1, we may assume that v j ∈ D ∗ sp , for otherwise the desired result holds. We now let C = { u ∈ N G [ v j ] | m ( u ) (cid:54) = 0 } , and consider two subcases. Case 1.1. C = ∅ .In this case we need to show that v j is either semipaired with a vertex in N T ( G ) [ F ( v j )] or with some sibling of F ( v j ) in D ∗ sp . Let v r be the vertex semipaired with v j in D ∗ sp and suppose neither v r / ∈ N T ( G ) [ F ( v j )] nor v r is asibling of F ( v j ) in T ( G ) . We note that in this case v r is at level l or l +1 in T ( G ) . Further, since C = ∅ , m ( u ) = 0 for all u ∈ N G [ v j ] . This implies that for all u ∈ N [ N [ v j ]] (that is, for all u such that d ( v j , u ) ≤ ), L ( u ) = 0 or . Thus, v r ∈ D i − implies v j ∈ D i − a contradiction. Hence, v r / ∈ D i − . If v r is at level l + 1 or a child of v j ,then using Lemma 4.4 and the fact that N i ( v r ) ⊆ N i ( v j ) we may conclude that D (cid:48) sp = ( D ∗ sp \ { v r } ) ∪ { F ( v j ) } with v j semipaired with F ( v j ) in D (cid:48) sp is the required minimum semi-PD-set. If v r is at level l and not a child of v j in T ( G ) , then v r is a child of some sibling v s of v j in T ( G ) such that v s v j ∈ E ( G ) . In this case N i [ v r ] ⊆ N i [ v s ] .Now if v s ∈ D ∗ sp then using Lemma 4.4, D (cid:48) sp = D ∗ sp \ { v r } ∪ { F ( v j ) } where v j is semipaired with F ( v j ) in D (cid:48) sp is the required minimum semi-PD-set. If v s / ∈ D ∗ sp , then using Lemma 4.4, D (cid:48) sp = ( D ∗ sp \ { v r } ) ∪ { v s } where v j is semipaired with v s in D (cid:48) sp is the required minimum semi-PD-set. Case 1.2. C (cid:54) = ∅ .Let k = min { b | v b ∈ C } and let m ( v k ) = r . By Lemma 4.4, k ≥ i . If v r is semipaired with v j in D ∗ sp thenthe result follows; otherwise, let v s and v t be the vertices semipaired with v j and v r , respectively, in D ∗ sp . Notethat m ( v k ) = r , and so L ( v r ) = 1 , that is, v r is not semipaired until the ( i − th − iteration. Hence, v t / ∈ D i − .Also by case . , either v t ∈ N T ( G ) [ v k ] or v t is a sibling of v k in T ( G ) .Suppose that v k is a child of v j in T ( G ) . In this case, v t is a child of either v j or v k in T ( G ) . If v t isa child of v k then N i ( v t ) ⊆ N i [ v j ] and if v t is a child of v j then N i [ v t ] ⊆ N i [ v j ] . Now, if N G ( v s ) ⊆ D ∗ sp ,then by Lemma 4.4 D (cid:48) sp = D ∗ sp \ { v s , v t } is semi-PD-set of smaller size, a contradiction. Hence there exist avertex u ∈ N G ( v s ) such that u / ∈ D ∗ sp and D (cid:48) sp = ( D ∗ sp \ { v t } ) ∪ { u } with v j and v s semipaired with v r and u ,respectively, is the required minimum semi-PD-set. Hence, we may assume that v k is not a child of v j in T ( G ) , forotherwise the desired result follows. So, v k can be one of the following: ( a ) a sibling of v j , ( b ) v j , or ( c ) F ( v j ) .We will consider these remaining cases under inclusion or exclusion of v s in D i − . Case 1.2.1 v s ∈ D i − .If L ( v s ) = 2 , then v j ∈ D i − , a contradiction. Hence, L ( v s ) = 1 . Thus, if v s ∈ D i − , then there exists avertex v s such that m ( v s ) = s , where v s = F ( v s ) and v s ∈ N G [ v j ] . If v s is at level l + 1 in T ( G ) , then v s isa child of v j , a contradiction noting that k = min { s | v s ∈ C } . If v s is at level l in T ( G ) , then v s is a sibling of v j in T ( G ) and s > k , implying that v k (cid:54) = F ( v j ) . If v s is at level l − , then v s = F ( v j ) hence, v k (cid:54) = F ( v j ) .Therefore, we may note that if v s ∈ D i − then v k (cid:54) = F ( v j ) . So v k may be a sibling of v j or v k = v j .Let v k is a sibling of v j . Now, v k v j ∈ E ( G ) and either s = j or v s v j ∈ E ( G ) . If v t is a child of v k , thenusing the fact that N i [ v t ] ⊆ N i [ v k ] , the set D (cid:48) sp = ( D ∗ sp \ { v t } ) ∪ { v k } with v j semipaired with v r and with v s semipaired with v k is the required minimum semi-PD-set. If v t is not a child of v k , then v t is either a sibling of v k or F ( v j ) . In both cases, we observe that d ( v s , v t ) ≤ . Hence, if we exchange the semipairs, that is, if we semipair v j and v r , and semipair v s and v t in the D ∗ sp , then D ∗ sp is the desired minimum semi-PD-set.Let v j = v k . If v t is a child of v k and N G ( v s ) ⊆ D ∗ sp then using Lemma 4.4 and the fact that N i [ v t ] ⊆ N i [ v k ] ,the set D (cid:48) sp = D ∗ sp \ { v s , v t } is a semi-PD-set of smaller size, a contradiction. Therefore, if v t is a child of v k , then there exists a vertex u ∈ N G ( v s ) such that u / ∈ D ∗ sp . In this case, using Lemma 4.4 and the fact that N i [ v t ] ⊆ N i [ v k ] , the set D (cid:48) sp = ( D ∗ sp \ { v t } ) ∪ { u } with v j semipaired with v r and with v s semipaired with u is the required minimum semi-PD-set. If v t is not a child of v k in T ( G ) , then observe that d ( v s , v t ) ≤ . Hence,if we exchange the semipairs, that is, we semipair v j and v r , and semipair v s and v t in the D ∗ sp , then D ∗ sp is thedesired minimum semi-PD-set. Case 1.2.2. v s / ∈ D i − .If s ≤ i or v s is a child of v j in T ( G ) and N G ( v t ) ⊆ D ∗ sp , then D (cid:48) sp = D ∗ sp \ { v s , v t } is a semi-PD-set ofsmaller size, a contradiction noting that N i ( v s ) ⊆ N i ( v j ) . Hence, if s ≤ i or v s is a child of v j in T ( G ) , thenthere exist a vertex u ∈ N G ( v t ) such that u / ∈ D ∗ sp and in this case D (cid:48) sp = ( D ∗ sp \ { v s } ) ∪ { u } with v j and v t semipaired with v r and u , respectively, is the required minimum semi-PD-set. Now we have, s > i and v s is not achild of v j , implying that in T ( G ) , the vertex v s may be one of the following: (i) a child of some sibling v (cid:48) s of v j in T ( G ) , (ii) v s is a sibling of v j , (iii) v s = F ( v j ) , (iv) v s is a sibling of F ( v j ) or (v) v s = F ( F ( v j )) .Suppose firstly that v k is a sibling of v j in T ( G ) . If v t is a child of v k , then using the fact that N i [ v t ] ⊆ N i [ v k ] ,the set D (cid:48) sp = ( D ∗ sp \ { v t } ) ∪ { v k } with v j semipaired with v r , and v s semipaired with v k is the required minimumsemi-PD-set. If v t is not a child of v k , then v t is either a sibling of v k or F ( v j ) . In both cases, we observe that d ( v s , v t ) ≤ . Hence, if we exchange the semipairs, that is, if we semipair v j and v r , and semipair v s and v t in the D ∗ sp , then D ∗ sp is the desired minimum semi-PD-set.Suppose secondly that v j = v k . If v t is a child of v k and N [ v s ] ⊆ D ∗ sp , then using the fact that N i [ v t ] ⊆ N i [ v k ] ,the set D (cid:48) sp = D ∗ sp \ { v s , v t } is semi-PD-set of smaller size, a contradiction. Therefore, if v t is a child of v k , then here exists a vertex u ∈ N ( v s ) such that u / ∈ D ∗ sp . In this case, using Lemma 4.4 and the fact that N i [ v t ] ⊆ N i [ v k ] ,the set D (cid:48) sp = ( D ∗ sp \ { v t } ) ∪ { u } with v j semipaired with v r , and with v s semipaired with u , is the requiredminimum semi-PD-set. If v t is not a child of v k in T ( G ) , then observe that d ( v s , v t ) ≤ . Hence, if we exchangethe semipairs, that is, if we semipair v j and v r , and semipair v s and v t in the D ∗ sp , then D ∗ sp is the desired minimumsemi-PD-set.Suppose next that v k = F ( v j ) . Let v s be a child of some sibling v s of v j in T ( G ) , implying that v s is achild of F ( v j ) . We note that as v k = F ( v j ) , this implies v t ∈ N T ( G ) ( F ( v j )) or v t is a sibling of F ( v j ) and v t F ( v j ) ∈ E ( G ) . Hence we may observe that d ( v s , v t ) ≤ . Therefore, if v s is a child of some sibling v s of v j in T ( G ) , then using Lemma 4.4 and the fact that N i [ v s ] ⊆ N i [ v s ] , the set D (cid:48) sp = ( D ∗ sp \ { v s } ) ∪ { v s } with v j semipaired with v r , and with v t semipaired with v s , is the required minimum semi-PD-set. If v s is not a child ofsome sibling v s of v j in T ( G ) , then we observe that d ( v s , v t ) ≤ . Hence, if we exchange the semipairs, that is,if we semipair v j and v r , and semipair v s and v t in the D ∗ sp , then D ∗ sp is the desired minimum semi-PD-set. Case 2. D ( v i ) (cid:54) = 0 and m ( v i ) = k (cid:54) = 0 .Using induction, there is minimum semi-PD-set D ∗ sp , such that D i − ⊆ D ∗ sp and v k is semipaired with avertex either in N T ( G ) [ v i ] or a sibling of v i . Using Observation 4.3, we note that for all vertices u ∈ N G [ v i ] \ { v k } either L ( u ) = 0 or and D ( v j ) = 1 for j ∈ [ i − by Lemma 4.4. Let v k be semipaired with v p . We note that v p / ∈ D i − . Case 2.1. L ( F ( v i )) = 0 .If v p = F ( v i ) , then the result follows. Let v p (cid:54) = F ( v i ) , implying that v p is a child of v i or a sibling of v i or p = i . If v p is child of v i then N i ( v p ) ⊆ N i [ F ( v i )] and if v p is a sibling of v i or p = i then we note that N i [ v p ] ⊆ N i [ F ( v i )] . Using Lemma 4.4, we can update D (cid:48) sp = D ∗ sp \ { v p } ∪ { F ( v i ) } with v k semipaired with F ( v i ) to get the required minimum semi-PD-set. Case 2.2. L ( F ( v i )) (cid:54) = 0 and L ( v i ) = 0 .In this case, F ( v i ) ∈ D i − and v p (cid:54) = F ( v i ) . If v p = v i in D ∗ sp , then D ∗ sp is the desired set. Now suppose v p (cid:54) = v i , implying that v p is either a child or a sibling of v i . Similar to the previous case, we note that if v p is childof v i then N i ( v p ) ⊆ N i [ F ( v i )] and if v p is a sibling of v i or p = i then N i [ v p ] ⊆ N i [ F ( v i )] . Using Lemma 4.4we can update D (cid:48) sp = ( D ∗ sp \ { v p } ) ∪ { v i } with v k semipaired with v i to get the required minimum semi-PD-set. Case 2.3. L ( F ( v i )) = L ( v i ) (cid:54) = 0 .In this case, F ( v i ) , v i ∈ D i − and neither F ( v i ) = v p nor v i = v p . Here, v p is either a child or a sibling of v i and similar to the previous cases, we note that N i [ v p ] ⊆ N i [ F ( v i )] ∪ N i [ v i ] . Since v p / ∈ D i − , we have L ( v p ) = 0 .Therefore, D ∗ sp with v k semipaired with v p is the desired minimum semi-PD-set. Case 3. D ( v n ) = 0 .Since D ( v n ) = 0 , L ( u ) = 0 for all u ∈ N G [ v n ] . Hence, no neighbor of v n is selected till ( n − th -iteration.Let v p be the vertex dominating v n in D ∗ sp and v q be the vertex semipaired with v p in D ∗ sp . If v q ∈ D n − ,then L ( v q ) = 1 . Hence, by Lemma 4.4, v q is at level or less in T ( G ) , a contradiction as D ( v n ) = 0 . If v q / ∈ D n − , then using the fact that L ( u ) = 0 for all u ∈ N G [ v n ] and by Lemma 4.4, we can state that D (cid:48) sp = D ∗ sp \ { v p , v q } ∪ { v n , u } where u ∈ N G ( v n ) is the desired minimum semi-PD-set.This completed the proof of Lemma 4.5.Now, we are ready to state the main result of this Section. Theorem 4.2.
Given a block graph G , a minimum semi-PD-set of G can be computed in O ( n + m ) -time.Proof. By Lemma 4.5, we claim that there is a minimum semi-PD-set D ∗ sp such that D n ⊆ D ∗ sp where D n isthe semi-PD-set returned by the Algorithm . This proves that the set D n returned by Algorithm is a minimumsemi-PD-set of G . Next, we analyze the complexity of computing D n for a given block graph G .By Theorem 4.1, given a block graph G = ( V, E ) , a BEO of vertices of G can be computed in O ( n + m ) -time, and the corresponding block tree can also be constructed in O ( n + m ) -time. Now, given a block tree T , wecan find the reverse of BFS ordering of T in O ( n + m ) -time. Also, all the computations in Algorithm can beperformed in O ( n + m ) -time. This proves that a minimum semi-PD-set of any block graph can be computed in O ( n + m ) -time. APX-completeness for Bounded Degree Graphs
In this section, we show that the M
INIMUM S EMIPAIRED D OMINAION problem is APX-complete for graphswith maximum degree . It is known that the M INIMUM S EMIPAIRED D OMINATION problem for a graph G withmaximum degree ∆ can be approximated with an approximation ratio of [10]. Hence the M INIMUM S EMIPAIRED D OMINATION problem is in APX for bounded degree graphs. To show APX-completeness, we usethe concept of L-reduction. First, we recall the definition of L-reduction.
Definition 5.1.
Given two NP optimization problems F and G and a polynomial time transformation f frominstances of F to instances of G , we say that f is an L-reduction if there are positive constants α and β such thatfor every instance x of F the following holds.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 we can in polynomial time finda solution y (cid:48) of x with m F ( x, y (cid:48) ) = c such that | opt F ( x ) − c | ≤ β | opt G ( f ( x )) − c | .To show the APX-completeness of a problem Π ∈ APX, it suffices to show that there is an L-reduction fromsome APX-complete problem to Π . We first show that the M
INIMUM S EMIPAIRED D OMINAION problem is APX-complete for graphs with max-imum degree . To show this result, we prove that the reduction given in the proof of Theorem in [10] is anL-reduction. So, we show an L-reduction from the M INIMUM V ERTEX C OVER PROBLEM for graphs with max-imum degree [1]. The M INIMUM V ERTEX C OVER problem is already known to be APX-complete for graphswith maximum degree . For a graph G = ( V, E ) , a set S ⊆ V is called a vertex cover of G if for every edge e = uv ∈ E , S ∩ { u, v } (cid:54) = ∅ . For a graph G , the M INIMUM V ERTEX C OVER problem is to find a vertex coverof G of minimum cardinality. Next, we present an L-reduction from the M INIMUM S EMIPAIRED D OMINATION problem for graphs with maximum degree to the M INIMUM S EMIPAIRED D OMINATION problem for graphswith maximum degree . Theorem 5.1.
The M INIMUM S EMIPAIRED D OMINATION problem is APX-complete for graphs with maximumdegree .Proof. The M
INIMUM S EMIPAIRED D OMINATION problem is in APX for graphs with maximum degree . Hence,to prove the APX-completeness, it is sufficient to give an L-reduction f , from the set of instances for the M INI - MUM V ERTEX C OVER PROBLEM for graphs with maximum degree , to the set of instances for the M INIMUM S EMIPAIRED D OMINATION problem for graphs with maximum degree .Given a graph G = ( V, E ) , where V = { v , v , . . . , v n } with d G ( v i ) ≤ for each i ∈ [ n ] and E = { e , e , . . . , e m } , we construct a graph G (cid:48) = ( V (cid:48) , E (cid:48) ) as follows:Let V k = { v ki | i ∈ [ n ] } , E k = { e kj | j ∈ [ m ] } for k ∈ [2] and X = { w i , x i , y i , z i | i ∈ [ n ] } . Define V ( G (cid:48) ) = V ∪ V ∪ E ∪ E ∪ X , and E ( G (cid:48) ) = { v i w i , v i w i , w i x i , x i y i , y i z i | i ∈ [ n ] } ∪ { v li e lj , v lk e lj | l ∈ [2] , j ∈ [ m ] and e j = v i v k ∈ E } . Fig. 4 illustrates the construction of G (cid:48) from G . Note that if the degree of a vertex in G isbounded by , then a vertex in G (cid:48) has degree at most .Next, we show that the above reduction is an L-reduction. The following claim is enough to complete theproof of the theorem. Claim 5.1. If V ∗ c denotes a minimum vertex cover of G and D ∗ sp denotes a minimum cardinality semi-PD-set of G (cid:48) ,and n denotes the number of vertices of G , then | D ∗ sp | = 2 | V ∗ c | + 2 n . Further, if D sp is an arbitrary semi-PD-setof G (cid:48) , then we can construct a vertex cover V c of G , such that | V c | − | V ∗ c | ≤ | D sp | − | D ∗ sp | .Proof. Let V ∗ c denotes a minimum vertex cover of G . Then, the set D sp = { v i , v i | v i ∈ V ∗ c } ∪ { w i , y i | i ∈ [ n ] } is a semi-PD-set of G (cid:48) which implies that | D sp | ≤ | V ∗ c | + 2 n . Hence, if D ∗ sp denotes a semi-PD-set of G (cid:48) ofminimum cardinality, then | D ∗ sp | ≤ | V ∗ c | + 2 n . b bb bbbb bbbb bbbb bbbb bbbbbbb bbb bbb bbb v v v v e e e e v v v v v v v v e e e e e e e e w x y z w w w x x x y y y z z z G G ′ Figure 4: An illustration of the construction of G (cid:48) from G in the proof of Theorem 5.1. Next, suppose that D sp is an arbitrary semi-PD-set of G (cid:48) . Note that | D sp ∩ { w i , x i , y i , z i }| ≥ for each i ∈ [ n ] . Hence, without loss of generality, we may assume that { w i , y i | i ∈ [ n ] } ⊆ D sp (with w i and y i paired in D sp ). Let D = D sp \ { w i , y i | i ∈ [ n ] } . We note that | D | = | D sp | − n . Let D = D ∩ ( V ∪ E ) and D = D ∩ ( V ∪ E ) . Renaming thesets if necessary, we may assume that | D | ≤ | D | , implying that | D | ≤ | D | / . In order to dominate a vertex e i ∈ E , either e i ∈ D sp or v j ∈ D sp where v j ∈ N G (cid:48) ( e i ) . If N G (cid:48) ( e i ) ∩ D sp = ∅ , then we update D as D = ( D \ { e i } ) ∪ { v j } for some v j ∈ N G (cid:48) ( e i ) . We note that the cardinality of the set D remains unchangedafter updating D for all such e i , and so, | D | ≤ | D | / | D sp | − n ) / . Also every vertex in E is nowdominated by some vertex in D . Therefore the set V c = { v i | v i ∈ D } is a vertex cover of G and | V c | = | D | ≤ ( | D sp | − n ) . Hence, | D sp | ≥ | V c | + 2 n . Now, if V ∗ c is a minimumvertex cover of G , then | D sp | ≥ | V ∗ c | + 2 n . This is true for every semi-PD-set, D sp , of G (cid:48) . In particular, if D ∗ sp is a minimum semi-PD-set of G (cid:48) , then we have | D ∗ sp | ≥ | V ∗ c | + 2 n . As observed earlier, | D ∗ sp | ≤ | V ∗ c | + 2 n .Consequently, | D ∗ sp | = 2 | V ∗ c | + 2 n . Further, | V c | − | V ∗ c | ≤
12 ( | D sp | − n ) −
12 ( | D ∗ sp | − n ) = 12 ( | D sp | − | D ∗ sp | ) . Since the maximum degree of G is and G is connected, n − ≤ m ≤ | V ∗ c | . Therefore, | D ∗ sp | = 2 | V ∗ c | +2 n ≤ | V ∗ c | + 2(3 | V ∗ c | + 1) ≤ | V ∗ c | + 2 ≤ | V ∗ c | . This proves that f is an L-reduction with α = 10 and β = .We observe that the graph G (cid:48) constructed in Theorem 5.1 is also a bipartite graph. Hence, as a corollary ofTheorem 5.1, we have the following result. Corollary 1.
The M INIMUM S EMIPAIRED D OMINATION problem is APX-complete for bipartite graphs withmaximum degree . Theorem 5.2.
The M INIMUM S EMIPAIRED D OMINATION problem is APX-complete for graphs with maximumdegree . roof. To prove the theorem, it is sufficient to describe an L-reduction h , from the M INIMUM S EMIPAIRED D OM - INATION for graphs with maximum degree to the M INIMUM S EMIPAIRED D OMINATION for graphs with maxi-mum degree .Given a graph G with maximum degree , we construct a graph G (cid:48) with maximum degree as follows: for avertex v ∈ V ( G ) with d G ( v ) = 4 , we split and transform v as illustrated in Fig. 5. For a vertex v with d G ( v ) ≤ ,we do not perform any transformation. b bbbb bb b b b b bbb b vu v v v v v v u u u u u u u b v Figure 5: Transformation of a vertex v ∈ V ( G ) with d G ( v ) = 4 . By construction, for every vertex v in G (cid:48) , we have d G (cid:48) ( v ) ≤ . Next, we show that the above reduction is anL-reduction. The following claim is enough to complete the proof of the theorem. Claim 5.2. If D ∗ sp and D (cid:48)∗ sp are minimum semi-PD-sets of G and G (cid:48) , respectively, and k denotes the number ofvertices of degree in G , then | D (cid:48)∗ sp | = | D ∗ sp | + 2 k . Further, if D (cid:48) sp is an arbitrary semi-PD-set of G (cid:48) , then we canconstruct a semi-PD-set D sp of G , such that | D sp | − | D ∗ sp | ≤ | D (cid:48) sp | − | D (cid:48)∗ sp | .Proof. Let D sp be a semi-PD-set of G . We construct a semi-PD-set D (cid:48) sp of G (cid:48) as follows:(1) If d G ( v ) ≤ , v ∈ D (cid:48) sp if and only if v ∈ D sp .(2) If d G ( v ) = 4 , then we proceed as follows.(2.1) If v ∈ D sp and v is semipaired with a vertex u ∈ N [ w ] such that w ∈ { u , u } , take v , v , v in D (cid:48) sp .(2.2) If v ∈ D sp and v is semipaired with a vertex u ∈ N [ w ] such that w ∈ { u , u } , take v , v , v in D (cid:48) sp .(2.3) If v / ∈ D sp and v is dominated by a vertex in the set { u , u } , take v , v in D (cid:48) sp .(2.4) If v / ∈ D sp and v is dominated by a vertex in the set { u , u } , take v , v in D (cid:48) sp .Let k be the number of vertices of degree in G . We observe that, D (cid:48) sp is a semi-PD-set of the transformed graph G (cid:48) and | D (cid:48) sp | = | D sp | + 2 k . Thus, | D (cid:48)∗ sp | ≤ | D ∗ sp | + 2 k , where D (cid:48)∗ sp and D ∗ sp are minimum semi-PD-sets of G (cid:48) and G , respectively.Conversely, let D (cid:48) sp be a semi-PD-set of G (cid:48) . Now we construct a semi-PD-set D sp of G from D (cid:48) sp . If d G ( v ) ≤ , then we will include v in D sp if and only if v ∈ D (cid:48) sp . If d G ( v ) = 4 , then v is transformed as shown in Fig. 5.For each vertex v ∈ V ( G ) of degree , let φ ( v ) = |{ v , v , . . . , v } ∩ D (cid:48) sp | . If the degree of v in G is , then weinclude v in D sp if and only if φ ( v ) ≥ . Without loss of generality, we may assume that, to dominate v either v ∈ D (cid:48) sp or v ∈ D (cid:48) sp . Also, v and v can be semipaired only with the vertices in the set { v , v , . . . , v } . Hence, φ ( v ) ≥ .We note that the vertices in the set { v , v , v , v , v } cannot be dominated by a vertex w ∈ D (cid:48) sp \{ v , v , . . . , v } .Hence, if φ ( v ) = 2 , then without loss of generality we may assume that either { v , v } ⊆ D (cid:48) sp and v issemipaired with v in D (cid:48) sp or { v , v } ⊆ D (cid:48) sp and v is semipaired with v in D (cid:48) sp . Note that in either case, V ( G (cid:48) ) \ { v , v , . . . , v } is dominated by D (cid:48) sp \ { v , v , . . . , v } .A vertex u ∈ D (cid:48) sp \ { v , v , . . . , v } can only be semipaired with a vertex in the set { v , v , v , v } in D (cid:48) sp .If φ ( v ) = 3 , then only one vertex in the set { v , v , v , v } is semipaired with a vertex w such that w ∈ D (cid:48) sp \{ v , v , . . . , v } . If w ∈ V ( G ) , then in D sp , the vertex v will be semipaired with vertex w . If w / ∈ G , that is, if w is obtained by splitting some vertex u of degree in G , then v will be semipaired with vertex u . uppose that φ ( v ) ≥ and more than two vertices in the set { v , v , v , v } are semipaired in D (cid:48) sp \{ v , v , . . . , v } .For simplicity, suppose { x, y } ⊆ D (cid:48) sp \ { v , v , . . . , v } where x and y are semipaired with some vertices in the set { v , v , v , v } . Note that V ( G (cid:48) ) \{ v , v , . . . , v } is dominated by ( D (cid:48) sp \{ v , v , . . . , v } ) ∪{ v } . If { x, y } ⊆ D sp ,then semipair x with v in D sp . Now it may be the case that y is not semipaired in D sp . If the vertex y ∈ D sp hasno partner in D sp and N G ( y ) ⊆ D sp , then remove y from D sp ; otherwise, we will include another vertex u / ∈ D sp which is at distance at most from y .The resulting set D sp is a semi-PD-set of G . We note that, | D sp | ≤ | D (cid:48) sp | − k . Thus, | D ∗ sp | ≤ | D (cid:48)∗ sp | − k and hence, | D (cid:48)∗ sp | = | D ∗ sp | + 2 k . Consequently, we have | D sp | − | D sp ∗ | ≤ | D (cid:48) sp | − | D (cid:48)∗ sp | .Finally, since G is a graph with maximum degree , for any dominating set D of G , we have | D | ≥ | V ( G ) | / .In particular, | D ∗ sp | ≥ | V ( G ) | / . Since k ≤ | V ( G ) | ≤ | D ∗ sp | , we have | D (cid:48)∗ sp | ≤ | D ∗ sp | + 2 k ≤ | D ∗ sp | . Hence, wemay conclude that h is an L-reduction from the M INIMUM S EMIPAIRED D OMINATION for graphs with maximumdegree to the M INIMUM S EMIPAIRED D OMINATION for graphs with maximum degree with α = 9 and β = 1 . The S
EMIPAIRED D OMINATION D ECISION problem is already known to be NP-complete for chordal graphs.In this paper, we study this problem for two important subclasses of chordal graphs: split graphs and block graphs.We show that the S
EMIPAIRED D OMINATION D ECISION problem remains NP-complete for split graphs, and wepropose a linear-time algorithm to compute a minimum cardinality semi-PD-set of a block graph. We also provethat the M
INIMUM S EMIPAIRED D OMINATION is APX-complete for graphs with maximum degree . It will beinteresting to study the complexity status of the problem for other important subclasses of chordal graphs, forexample strongly chordal graphs, doubly chordal graphs, undirected path graphs etc. References [1] P. Alimonti and V. Kann. Hardness of approximating problems on cubic graphs. Theor. Comput. Sci., 237(2000) 123 ˝U134.[2] T. W. Haynes, M. A. Henning. Perfect graphs involving semitotal and semipaired domination. J. Comb.Optim., 36 (2018) 416–433.[3] T. W. Haynes, M. A. Henning. Semipaired domination in graphs. J. Combin. Math. Combin. Comput., 104(2018) 93–109.[4] T. W. Haynes and M. A. Henning. Graphs with large semipaired domination number. Discuss. Math. GraphTheory (2019), 659–671.[5] T. W. Haynes and P. J. Slater. Paired domination in graphs. Networks, 32 (1998) 199–206.[6] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater. Fundamentals of Domination in Graphs, volume 208. MarcelDekker Inc., New York, 1998.[7] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater. Domination in graphs: Advanced topics, volume 209. MarcelDekker Inc., New York, 1998.[8] M. A. Henning and P. Kaemawichanurat. Semipaired Domination in claw-free cubic graphs. Graphs Combin.,34 (2018) 819–844.[9] M. A. Henning and P. Kaemawichanurat. Semipaired domination in maximal outerplanar graphs. J Comb.Optim., 38 (2019) 911–926.[10] M. A. Henning, A. Pandey, and V. Tripathi, Complexity and Algorithms for Semipaired Domination inGraphs. In IWOCA 2019, Lecture Notes in Comput. Sci., 11638 (2019) 278–289.
11] M. A. Henning and A. Pandey. Algorithmic aspects of semitotal domination in graphs. Theor. Comput. Sci.,766 (2019) 46–57.[12] Lie Chen, Changhong Lu, and Zhenbing Zeng. Labelling Algorithms for paired domination problems inblock and interval graphs. J. Comb. Optim., 19 (2010) 457-470.[13] W. Desormeaux and M. A. Henning. Paired domination in graphs: A survey and recent results. Util Math, 94(2014) 101-166.[14] W. Zhuang, Guoliang Hao. Domination versus semipaired domination in trees, Quaestiones Mathematicae(2019) DOI: 10.2989/16073606.2019.164156611] M. A. Henning and A. Pandey. Algorithmic aspects of semitotal domination in graphs. Theor. Comput. Sci.,766 (2019) 46–57.[12] Lie Chen, Changhong Lu, and Zhenbing Zeng. Labelling Algorithms for paired domination problems inblock and interval graphs. J. Comb. Optim., 19 (2010) 457-470.[13] W. Desormeaux and M. A. Henning. Paired domination in graphs: A survey and recent results. Util Math, 94(2014) 101-166.[14] W. Zhuang, Guoliang Hao. Domination versus semipaired domination in trees, Quaestiones Mathematicae(2019) DOI: 10.2989/16073606.2019.1641566