Complexity and Algorithms for Semipaired Domination in Graphs

04/01/2019 ∙ by Michael A. Henning, et al. ∙ University of Johannesburg Indian Institute of Technology Ropar 0

For a graph G=(V,E) with no isolated vertices, a set D⊆ V is called a semipaired dominating set of G if (i) D is a dominating set of G, and (ii) D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by γ_pr2(G). The Minimum Semipaired Domination problem is to find a semipaired dominating set of G of cardinality γ_pr2(G). In this paper, we initiate the algorithmic study of the Minimum Semipaired Domination problem. We show that the decision version of the Minimum Semipaired Domination problem is NP-complete for bipartite graphs and split graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs and trees. We also propose a 1+(2Δ+2)-approximation algorithm for the Minimum Semipaired Domination problem, where Δ denote the maximum degree of the graph and show that the Minimum Semipaired Domination problem cannot be approximated within (1-ϵ) |V| for any ϵ > 0 unless NP ⊆ DTIME(|V|^O(|V|)).

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1 Introduction

A dominating set in a graph is a set of vertices of such that every vertex in is adjacent to at least one vertex in . The domination number of , denoted by , is the minimum cardinality of a dominating set of . The Minimum Domination problem is to find a dominating set of cardinality . More thorough treatment of domination, can be found in the books [6, 7]. A dominating set is called a paired dominating set if contains a perfect matching. The paired domination number of , denoted by is the minimum cardinality of paired dominating set of . The concept of paired domination was introduced by Haynes and Slater in [11].

A relaxed form of paired domination called semipaired domination was introduced by Haynes and Henning [8] and studied further in [12, 9, 10]. A set of vertices in a graph with no isolated vertices is a semipaired dominating set, abbreviated a semi-PD-set, of if is a dominating set of and can be partitioned into -element subsets such that the vertices in each -element set are at distance at most . In other words, the vertices in the dominating set can be partitioned into -element subsets such that if is a -set, then the distance between and is either or . We say that and are semipaired. The semipaired domination number of , denoted by , is the minimum cardinality of a semi-PD-set of . Since every paired dominating set is a semi-PD-set, and every semi-PD-set is a dominating set, we have the following observation.

Observation 1.1.

([8]) For every isolate-free graph , .

By Observation 1.1, the semipaired domination number is squeezed between two fundamental domination parameters, namely the domination number and the paired domination number.

More formally, the minimum semipaired domination problem and its decision version are defined as follows:

Minimum Semipaired Domination problem (MSPDP)

  1. Instance: A graph .

  2. Solution: A semi-PD-set of .

  3. Measure: Cardinality of the set .

Semipaired Domination Decision problem (SPDDP)

  1. Instance: A graph and a positive integer .

  2. Question: Does there exist a semi-PD-set in such that ?

In this paper, we initiate the algorithmic study of the semipaired domination problem. The main contributions of the paper are summarized below. In Section 2, we discuss some definitions and notations. In Section 3, we discuss the difference between the complexity of paired domiantion and semipaired domination in graphs. In Section 4, we show that the Semipaired Domination Decision problem is NP-complete for bipartite and split graphs. In Section 5 and Section 6, we propose a linear-time algorithms to solve the Minimum Semipaired Domination problem in interval graphs and trees respectively. In Section 7, we propose an approximation algorithm for the Minimum Semipaired Domination problem in general graphs. In Section 8, we discuss an approximation hardness result. Finally, Section 9, concludes the paper.

2 Terminology and Notation

For notation and graph theory terminology, we in general follow [13]. Specifically, let be a graph with vertex set and edge set , and let be a vertex in . The open neighborhood of is the set and the closed neighborhood of is . Thus, a set of vertices in is a dominating set of if for every vertex , while is a total dominating set of if for every vertex . The distance between two vertices and in a connected graph , denoted by , is the length of a shortest -path in . If the graph is clear from the context, we omit it in the above expressions. We write , and rather than , and , respectively.

For a set , the subgraph induced by is denoted by . If , where , is a complete subgraph of , then is a clique of . A set is an independent set if has no edge. A graph is chordal if every cycle in of length at least four has a chord, that is, an edge joining two non-consecutive vertices of the cycle. A chordal graph is a split graph if can be partitioned into two sets and such that is a clique and is an independent set. A vertex is a simplicial vertex of if is a clique of . An ordering is a perfect elimination ordering (PEO) of vertices of if is a simplicial vertex of for all , . Fulkerson and Gross [4] characterized chordal graphs, and showed that a graph is chordal if and only if it has a PEO. A graph is bipartite if can be partitioned into two disjoint sets and such that every edge of joins a vertex in to a vertex in , and such a partition of is called a bipartition of . Further, we denote such a bipartite graph by . A graph is an interval graph if there exists a one-to-one correspondence between its vertex set and a family of closed intervals in the real line, such that two vertices are adjacent if and only if their corresponding intervals intersect. Such a family of intervals is called an interval model of a graph.

In the rest of the paper, all graphs considered are simple connected graphs with at least two vertices, unless otherwise mentioned specifically. We use the standard notation . For most of the approximation related terminologies, we refer to [1, 14].

3 Complexity difference between paired domination and semipaired domination

In this section, we make an observation on complexity difference between paired domination and semipaired domination. We show that the decision version of the Minimum paired domination problem is NP-complete for GP graphs, but the Minimum Semipaired Domination problem is easily solvable for GP graphs. The class of GP graphs was introduced by Henning and Pandey in [15]. Below we recall the definition of GP graphs.

Definition 3.1 (Gp-graph).

A graph is called a GP-graph if it can be obtained from a general connected graph where , by adding a path of length  to every vertex of . Formally, and .

Theorem 3.1.

If is a GP-graph, then .

Lemma 3.1.

If is a GP-graph constructed from a graph as in Definition 3.1, then has a paired dominating set of cardinality , if and only if has a semi-PD-set of cardinality .

Since the decision version of the Minimum Paired Domination problem is known to be NP-complete for general graphs [11], the following theorem follows directly from Lemma 3.1.

Theorem 3.2.

The decision version of the Minimum Paired Domination problem is NP-complete for GP-graphs.

4 NP-completeness Results

In this section, we study the NP-completeness of the Semipaired Domination Decision problem. We show that the Semipaired Domination Decision problem is NP-complete for bipartite graphs and split graphs.

4.1 NP-completeness proof for bipartite graphs

Theorem 4.1.

The Semipaired Domination Decision problem is NP-complete for bipartite graphs.

Proof.

Clearly, the Semipaired Domination Decision problem is in NP for bipartite graphs. To show the hardness, we give a polynomial reduction from the Minimum Vertex Cover problem. Given a non-trivial graph , where and , we construct a graph in the following way:

Let and for . Also assume that , , , and .

Now define ,
and , are endpoints of edge in . Fig. 2 illustrates the construction of from .

Figure 1: An illustration of the construction of from in the proof of Theorem 4.1.

Note that the set is an independent set in . Also, the set is an independent set in . Since , the graph is a bipartite graph. Now to complete the proof, it suffices for us to prove the following claim:

Claim 4.1.

The graph has a vertex cover of cardinality at most  if and only if the graph has a semi-PD-set of cardinality at most .

Proof.

Let be a vertex cover of of cardinality . Then is a semi-PD-set of of cardinality .

Conversely, suppose that has a semi-PD-set of cardinality at most . Note that for each . Hence, without loss of generality, we may assume that , where and are semipaired. Hence . Let . Without loss of generality, we may also assume that . Now, if for some , and none of its neighbors belongs to , then must be semipaired with some vertex where , and also there must exists a vertex which is a common neighbor of and . In this case, we replace the vertex in the set with the vertex and so where and are semipaired. We do this for each vertex where with none of its neighbors in the set . For the resulting set , and every vertex has a neighbor in . The set is a vertex cover of of cardinality at most . This completes the proof of the claim. ∎

Hence, the theorem is proved. ∎

4.2 NP-completeness result for split graphs

Theorem 4.2.

The Semipaired Domination Decision problem is NP-complete for split graphs.

Proof.

Clearly, the Semipaired Domination Decision problem is in NP. To show the hardness, we give a polynomial time reduction from the Domination Decision problem, which is well known NP-complete problem. Given a non-trivial graph , where and , we construct a split graph as follows:

Let and for . Now define , and and . Note that the set is a clique in and the set is an independent set in . Since , the constructed graph is a split graph. Fig. 2 illustrates the construction of from .

Figure 2: An illustration to the construction of from in the proof of Theorem 4.2.

Now, to complete the proof of the theorem, we only need to prove the following claim.

Claim 4.2.

has a dominating set of cardinality if and only if has a semi-PD-set of size cardinality .

Proof.

Let be a dominating set of size atmost of . Then is a semi-PD-set of of size atmost .

Conversely, suppose that has a semi-PD-set of cardinality at most . Let and . Then either or . Without loss of generality, let us assume that . Note that if and none of neighbors belong to then we replace by some of its neighbor in the set . So, we may assume that . Now the set is a dominating set of of size atmost . Hence, the result follows. ∎

Hence, the theorem is proved. ∎

5 Algorithm for Interval Graphs

In this section, we present a linear-time algorithm to compute a minimum cardinality semi-PD-set of an interval graph.

A linear time recognition algorithm exists for interval graphs, and for an interval graph an interval family can also be constructed in linear time [2, 5]. Let be an interval graph and be its interval model. For a vertex , let be the corresponding interval. Let and denote the left and right end points of the interval . Without loss of generality, we may assume that no two intervals share a common end point. Let be the left end ordering of vertices of , that is, whenever . Now we first prove the following lemmas.

Lemma 5.1.

Let be the left end ordering of vertices of . If for , then for every .

Proof.

The proof directly follows from the left end ordering of vertices of . ∎

Define the set , for each .

Lemma 5.2.

If is a connected interval graph, then is also connected.

Proof.

The proof can easily be done using induction on . ∎

Let be the least index vertex adjacent to , that is, if , then . In particular, we define . Let , where and . In particular, if does not exist, we assume that . Let and denote a semi-PD-set of of minimum cardinality. Recall that we only consider connected graphs with at least two vertices.

Lemma 5.3.

For , if , then .

Proof.

Note that every vertex in is dominated by , and . Hence, . ∎

Lemma 5.4.

For , if , and , then .

Proof.

Note that every vertex in is dominated by some vertex in the set , and . Hence, . ∎

Lemma 5.5.

For , let , . If every vertex where , is adjacent to at least one vertex in the set , then the following holds:
(a) .
(b) is semipaired with in .
(c) .

Proof.

(a) To dominate , either or , where and . If and is semipaired with some vertex , then , and . Hence, we can update the set as and semipair with . This proves that .

If also belongs to , then we are done. Otherwise, if is semipaired with (where ), then . Also, . In that case, we can update the set as . Hence, .

(b) Suppose . If is semipaired with in , then we are done. Otherwise, if is not semipaired with , assume that is semipaired with and is semipaired with . Note that must be greater than , and . Therefore, the set also dominates all the vertices of .

Suppose that . In this case, is a semi-PD-set of where and are semipaired. This contradicts the fact that is a semi-PD-set of of minimum cardinality. Hence, .

Let . Now update the set as follows: remove from , add in the set , semipair with and with . Clearly, the updated set is also a semi-PD-set of of minimum cardinality. This proves that there always exists a semi-PD-set of such that , and is semipaired with in .

(c) We know that . We need to show that , that is, there is no other vertex from the set belongs to . Suppose, to the contrary, that there does not exist any for which . So, for each , . Consider a set for which is minimum.

Let . Also, assume that , where and . Also, assume that is semipaired with in . Now consider the following two cases.

Case 1. . If , then if, some vertex of the set is dominated by or , then that vertex is also dominated by . In that case, is also a semi-PD-set of , which is a contradiction. If and , then also is a semi-PD-set of , which is again a contradiction. Hence, and . Suppose . Then, update the set as . Note that is still a semi-PD-set of of minimum cardinality, and , a contradiction.

Case 2. . If , then the updated set is also a semi-PD-set of of minimum cardinality. If and , then the updated set is also a semi-PD-set of , a contradiction. If and , let . Then, update as . Note that is still a semi-PD-set of of minimum cardinality, and , a contradiction.

Since both Case 1 and Case 2 produce a contradiction, there exists a semi-PD-set of of minimum cardinality, for which the set contains only and . ∎

Lemma 5.6.

For , let , . If every vertex where , is adjacent to at least one vertex in the set , then the following holds.
(a) if .
(b) if .
(c) if with .

Proof.

(a) Clearly .

(b) From Lemma 5.5, we know that . Also, other than , all vertices are dominated by the set . Hence, .

(c) Clearly is a semi-PD-set of . Hence . We also know that there exists a semi-PD-set of of minimum cardinality such that (where and are semipaired in ). Hence . Also, dominates the set , implying that the set is dominated by the vertices in . Hence, the set is semi-PD-set of . Therefore, . This proves that . Hence, . ∎

Lemma 5.7.

For , let , , and . Let and (assume that such a exists). Let . Then, the following holds.
(a) .
(b) is semipaired with in .
(c) .

Proof.

(a) First we show that . Suppose . Let be the vertex dominating in . Note that and . Let be the vertex semipaired with in . Since , any vertex which is within distance  from is also within distance  from . We can update as with semipaired with . Hence, contains . Similarly, we can show that also contains . So, .

(b) If is semipaired with in , then we are done. Suppose, to the contrary, that is not semipaired with in . So, assume that is semipaired with and is semipaired with in . We consider the four cases based on the values of the indices and .

Case 1. and . Here, . Hence, the set is also a semi-PD-set of , a contradiction.

Case 2. and . Since the distance between and is at most , . If and , then . So, in the set , can be semipaired with , and can be semipaired with .

Case 3. and . Here, . If , then the set is also a semi-PD-set of , a contradiction. If , let . Then update as , and semipair with and with .

Case 4. and . Since the distance between and is at most , . Also . If , then the set is also a semi-PD-set of , a contradiction. If , let . Then update as , and semipair with and with .

By the above four cases, there always exists a semi-PD-set of of minimum cardinality such that is semipaired with in . This completes the proof of part (b).

(c) The proof is similar to the proof of Lemma 5.5(c), and hence is omitted. ∎

Lemma 5.8.

For , let , , and . Let and (assume that such a exists). Let . Then, the following holds.
(a) if .
(b) if .
(c) if with .

Proof.

The proof is similar to the proof of Lemma 5.6, and hence is omitted. ∎

Based on above lemmas, we present an algorithm to compute a minimum semi-PD-set of an interval graph.

Input: An interval graph with a left end ordering of vertices of .
Output: A semi-PD-set of of minimum cardinality.
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