A set $S \subseteq V(G)$  is a semitotal dominating set  of a graph $G$ if‎

 ‎it is a dominating set of $G$ and‎

‎every vertex in $S$ is within distance 2 of another vertex of $S$‎. ‎The‎

‎semitotal domination number $\gamma_{t2}(G)$ is the minimum‎

‎cardinality of a semitotal dominating set of $G$‎.

‎We show that the semitotal domination problem is‎

‎APX-complete for bounded-degree graphs‎, ‎and  the semitotal domination problem in any graph  of maximum degree $\Delta$ can be approximated with an approximation‎

‎ratio of $2+\ln(\Delta-1)$

Pu Wu

Zehui Shao

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CCOCommun. Comb. Optim.c© 2018 Azarbaijan Shahid Madani University. All rights reserved.Communications in Combinatorics and OptimizationVol. 3 No. 2, 2018 pp.143-150DOI: 10.22049/CCO.2018.25987.1065Complexity and approximation ratio of semitotal dominationin graphsZehui Shao and Pu WuInstitute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, Chinazshao@gzhu.edu.cnReceived: 29 July 2017; Accepted: 3 June 2018Published Online: 5 June 2018Communicated by Alireza Ghaffari-HadighehAbstract: A set S ⊆ V (G) is a semitotal dominating set of a graph G if it is adominating set of G and every vertex in S is within distance 2 of another vertex of S.The semitotal domination number γt2(G) is the minimum cardinality of a semitotaldominating set of G. We show that the semitotal domination problem is APX-completefor bounded-degree graphs, and the semitotal domination problem in any graph ofmaximum degree ∆ can be approximated with an approximation ratio of 2+ln(∆−1).Keywords: semitotal domination, APX-complete, NP-completeAMS Subject classification: 05C691. Terminology and introductionIn this paper, we shall only consider graphs without multiple edges or loops or isolatedvertices. For a graph G, S ⊆ V (G), v ∈ V (G), the open neighborhood of v in S isdenoted by NS(v) (or simply N(v)), i.e. NS(v) = {u : uv ∈ E(G), u ∈ S}.Domination and its variations in graphs have attracted considerable attention [3, 5, 6].A set S ⊆ V (G) is a dominating set of a graph G if every vertex in V (G) \ S isadjacent to a vertex in S. The domination number γ(G) is the minimum cardinalityof a dominating set of G. A set S ⊆ V (G) is a semitotal dominating set of a graphG if it is a dominating set of G and every vertex in S is within distance 2 of anothervertex of S. The semitotal domination number γt2(G) is the minimum cardinality ofa semitotal dominating set of G.The semitotal domination problem consists of finding the semitotal domination num-ber of a graph G. It has been proved to be NP-complete and was claimed that there144 Complexity and approximation ratio of semitotal domination in graphsis a linear-time algorithm for trees [4]. Henning studied the semitotal dominationin cubic claw-free graphs and proposed a conjecture, and soon the conjecture wasconfirmed in [8]. In this paper, we continue studying the complexity of the semitotaldomination problem and extend these studies by investigating the approximationhardness of the semitotal domination problem in graphs.2. NP-completeness of semitotal dominationGoddard et al. proved that the semitotal domination problem is NP-complete forgeneral graphs, where the semitotal domination problem is stated as follows.Semitotal Domination ProblemInput: A graph G, and an integer k.Question: Is there a semitotal dominating set of G with cardinalityat most k ?We show that it is NP-complete for bipartite, or chordal, or planar graphs via reduc-tion from the dominating set problem.Theorem 1. The semitotal domination problem is NP-complete for bipartite, or chordal,or planar graphs.Proof. It is clear the semitotal domination problem is in NP, since it is easy to verifya yes instance of the semitotal domination problem in polynomial time. Now let usshow how to transform any instance (G, k) of DOM into an instance (G′, k′) of thesemitotal domination problem so that G has a dominating set of order k if and onlyif G′ has a semitotal dominating set of order k′.Let G be an arbitrary graph, we construct a graph G′ as follows. For each vertexv ∈ V (G), we build a star Kv1,4 = {wv,1, wv,2, wv,3, wv,4} centered at wv,1, and add thestar Kv1,4 and connect wv,1 to v. That is to say, V (G′) = V (G)∪{wv,1, wv,2, wv,3, wv,4 :v ∈ V (G)}, and E(G′) = E(G) ∪ {wv,1wv,2, wv,1wv,3, wv,1wv,4, wv,1v : v ∈ V (G)}.Suppose that G has a dominating set of D, then we have D′ = D∪{wv,1 : v ∈ V (G)}is a semitotal dominating set of G′. It can be seen that |D′| = |D|+ |V (G)|.Conversely, suppose that G′ has a semitotal dominating set D′. Then we can obtaina semitotal dominating set D′′ such that |D′′| ≤ |D′| and D′′ ⊇ {wv,1 : v ∈ V (G)}and D′′ ∩ {wv,2, wv,3, wv,4 : v ∈ V (G)} = ∅. Now, we claim that D = D′′ \ {wv,1 : v ∈V (G)} is a dominating set of G. It can be seen that |D| = |D′′| − |V (G)|.Since G is bipartite (resp. chordal, planar), G′ is also bipartite (resp. chordal, planar).Note that the dominating set problem is NP-complete for bipartite, or chordal, orplanar graphs, so the semitotal dominating set problem is also NP-complete for suchgraphs.Z. Shao and P. Wu 1453. APX-completeness of semitotal dominationThe notation of L-reduction can be found in [1, 2, 7]. Given two NP optimizationproblems G1 and G2 and a polynomial time transformation h from instances of G1 toinstances of G2, h is said to be an L-reduction if there are positive constants α andβ such that for every instance x of G1, we have(1) optG1(h(x)) ≤ α · optG2(x);(2) for every feasible solution y of h(x) with objective value mG(h(x), y) = c2 wecan in polynomial time find a solution y of x with mG1(x, y) = c1 such that|optG1(x)− c1| ≤ β · |optG2(h(x))− c2|.To show that a problem P ∈ APX is APX-complete, we need to show that there isan L-reduction from some APX-complete problem to P. The following problem wasproved to be APX-complete (see [2]):MIN DOM SET-BInput: A graph G = (V,E) with degree at most B.Solution: A dominating set S of G.Measure: Cardinality of the dominating set S.Now we consider the following problem with B ∈ {3, 4}:SEMITOTAL DOM-BInput: A graph G = (V,E) with degree at most B.Solution: A semitotal dominating set S of G.Measure: Cardinality of the semitotal dominating set S.Theorem 2.i) SEMITOTAL DOM-4 is APX-complete;ii) SEMITOTAL DOM-3 is APX-complete.Proof. i) It is clear that SEMITOTAL DOM-4 ∈ APX, and so we just have to showSEMITOTAL DOM-4 is APX-hard. We will construct an L-reduction f from MINDOM-3 for cubic graphs to SEMITOTAL DOM-4 for graphs with maximum degree4. Given a cubic graph G, we construct a graph G′ with maximum degree 4 in thefollowing way. For each vertex v with N(v) = {a, b, c}, we split the vertex v andtransform to the gadget depicted in Fig. 1 (b).146 Complexity and approximation ratio of semitotal domination in graphsa b cv1v2v3v4v5v6abcduabcdu1 u2u3u4u5acvbCurveLet D is a dominating set of G, we construct a vertex set TD of G′ in the followingway:• If v ∈ D, then v2, v4, v6 are put to TD.• If v /∈ D, then we have that one of {a, b, c} is in D and thus v2, v6 are put to TD.It can be checked that TD is a semitotal dominating set of G′ and |TD| = |D| +2|V (G)|. In particular, we have |TD∗| ≤ |D∗| + 2|V (G)|, where TD∗ is a minimumsemitotal dominating set of G′ and D∗ is a minimum dominating set of G. It iswell known that γ(G) ≥ |V (G)|∆+1 , so we have |V (G)| ≤ 5|D∗|. Therefore, we have|TD∗| ≤ |D∗|+ 2|V (G)| ≤ 11|D∗|.Let TD′ be a semitotal dominating set of G′. We construct a vertex set D′ of Gas follows. Let T (v) = {v1, v2, v3, v4, v5, v6} for any v ∈ V (G) and s(v) = |T (v) ∩TD′|. If s(v) = 3, we put v to D′. We claim that D′ is a dominating set of G.|D′| ≤ |TD′| − 2|V (G)|. In particular, we have |D∗| ≤ |TD∗| − 2|V (G)| and thus|D∗| = |TD∗| − 2|V (G)|. In addition, |D| − |D∗| ≤ |TD′| − |TD∗|. As a result, f isan L-reduction with α = 11 and β = 1.ii) It is clear that SEMITOTAL DOM-3 ∈ APX, and so we just have to show SEMI-TOTAL DOM-3 is APX-hard. Since we have shown that SEMITOTAL DOM-4is APX-complete, we now consider the following L-reduction g from SEMITOTALDOM-4 to SEMITOTAL DOM-3.Given a graph G with maximum degree 4, we construct a graph G′ with maximumdegree 3 in the following way. For each vertex u with degree 4 and N(u) = {a, b, c, d},we split the vertex u and transform to the gadget depicted in Fig. 1 (a). Let TD1 bea semitotal dominating set of G, we construct a vertex set TD2 of G′ in the followingway:• If u ∈ TD1 and d(u) ≤ 3, then u is put to TD2.• If u ∈ TD1 and d(u) = 4, then u1, u3, u5 are put to TD2.• If u /∈ TD1 and d(u) = 4, then u2, u3 are put to TD2.It can be checked that TD2 is a semitotal dominating set of G′ and |TD2| = |TD1|+2k, where k is the number of vertices with degree 4 in G. In particular, |TD2∗| =Z. Shao and P. Wu 147|TD1∗| + 2k, where TD1∗ is a minimum semitotal dominating set of G and TD2∗is a minimum semitotal dominating set of G′. Since ∆(G) ≤ 4, we have 5|TD1| ≥|V (G)| ≥ k, and so |TD2| ≤ |TD1|+ 2k ≤ 11|TD1|.Let TD2 be a semitotal dominating set of G′. We construct a vertex set TD1 ofG as follows. For any u ∈ V (G) with d(u) = 4, let T (u) = {u1, u2, u3, u4, u5}and s(u) = |T (u) ∩ TD2|. For any u ∈ V (G) with d(u) ≤ 3, let T (u) = {u} ands(u) = |T (u) ∩ TD2|. If s(u) = 3 or s(u) = 1, we put u to TD1. We claim that TD1is a semitotal dominating set of G. |TD1| ≤ |TD2| − 2k, where k is the number ofvertices with degree 4 in G. In particular, we have |TD1∗| ≤ |TD2∗| − 2k and thus|TD1∗| = |TD2∗| − 2k In addition, |TD1| − |TD1∗| ≤ |TD2| − |TD2∗|. As a result, gis an L-reduction with α = 11 and β = 1.4. Approximation ratio of semitotal dominationGiven a graph G, let v ∈ V (G) and A be a family of subset of V (G), we defineF(A, v) = {S : S ∩N [v] 6= ∅, S ∈ A} and f(A, v) = |F(A, v)|.Algorithm 1: GreedySemiTotalDom(G);Output: D;begin1: A ← {{v1}, {v2}, · · · , {vn}};2: B ← ∅;3: D ← ∅;4: i← 0;5: Ai ← A;6: while(A 6= ∅)7: begin8: find a vertex v ∈ V (G) \D which maximizes f(A, v);9: T ← {S|S ∈ A, S ∩N [v] 6= ∅};10: A ← A \ T ;11: if (∀b ∈ B, b ∩N [v] = ∅)12: A ← A∪ {N [v]};13: endif14: B ← B ∪ {N [v]};15: D ← D ∪ {v};16: i← i+ 1;17: Ai ← A;18: end19: g ← i;20:end.Remark: Although it seems that Ai is not used in Algorithm 1, it will be usedin the analysis of the approximation ratio of Algorithm 1 for finding the semitotal148 Complexity and approximation ratio of semitotal domination in graphsdomination number of a graph in Theorem 3.Theorem 3. The SEMITOTAL DOM in any graph G = (V,E) of maximum degree∆(≥ 2) can be approximated with an approximation ratio of 2 + ln(∆− 1).Proof. We proceed with some claims.Claim 1. |Ai+1| ≤ |Ai| − 1 and Algorithm 1 can terminate within finite steps.Proof. If ∃b ∈ B such that b ∩ N [v] 6= ∅, then we have T 6= ∅ and thus |Ai+1| ≤|A| − |T | ≤ |Ai| − 1.If ∀b ∈ B, b ∩N [v] = ∅, then ∀v′ ∈ N [v] we have v′ ∈ Ai. Since |N [v]| ≥ 2, we have|T | ≥ 2 and thus |Ai+1| ≤ |Ai| − |T |+ 1 ≤ |Ai| − 1. Therefore, we have Algorithm 1can terminate within finite steps.Claim 2. D is a semitotal dominating set of G.Proof. Firstly, we have ∀v ∈ V (G), there exist v′ ∈ D such that v ∈ N [v′]. Other-wise, {v} /∈ T for any iteration of Algorithm 1 (see lines 9 and 10), since {v} ∈ A0we have {v} ∈ Ag, a contradiction.Secondly, we have ∀v′ ∈ D, ∃v 6= v′ such that v ∈ D and v′ ∈ N [v]. Otherwise, weassume the vertex v′ is selected at the i-th iteration of Algorithm 1, then N [v′] ∈Ai+1. Since Ag = ∅, we have there exists a vertex v such that N [v] ∩ N [v′] 6= ∅, acontradiction.Claim 3. Algorithm 1 has approximation ratio of 2 + ln(∆− 1).Proof. Let m be the semitotal domination number of G and U = {u1, u2, · · · , um}be a semitotal dominating set of G with |U | = m. If |A0| ≤ 2m, we have Claim 3holds. Now we only need to consider the case |A0| > 2m.Note that g equals |D| which is the output of Algorithm 1, we will show that g ≤m(2 + ln(∆− 1)).Firstly, we show that |Ai|−|Ai+1| ≥ |Ai|m −1 if |Ai| ≥ 2m. Since U = {u1, u2, · · · , um}is a semitotal dominating set of G, we have there exist a j such thatf(Ai, uj) ≥|Ai|m> 1.Since f(Ai, uj) is an integer, we have |Ai|m ≥ 2 and uj /∈ Di. Now we havem|Ai| −m|Ai+1| ≥ |Ai| −m,(m− 1)|Ai| ≥ m|Ai+1| −m.Z. Shao and P. Wu 149Consequently,|Ai+1| ≤ (1−1m)|Ai|+ 1,≤ (1− 1m)(1− 1m|Ai−1|+ 1) + 1,≤ 1 +i∑j=1(1− 1m)j + (1− 1m)i+1|A0|.Since |A0| = n, we have|Ai+1| ≤ m(1− (1− 1m)i+1)+ (1− 1m)i+1n. (1)Since |A0| > 2m, we have there exists an integer i such that |Ai| ≥ 2m and |Ai+1| <2m. By inequality (1), we have2m ≤ |Ai| ≤ m(1− (1− 1m)i)+ (1− 1m)in, (2)and thus2 ≤(1− (1− 1m)i)+ (1− 1m)inm. (3)Since nm ≤ ∆ and (1−1m )i ≤ e− im , we havei ≤ m(ln(∆− 1)). (4)Since |Ai+1| < 2m, we have g − (i + 1) < 2m. Since g − i is an integer, we haveg ≤ i+ 2m ≤ m(2 + ln(∆− 1)).AcknowledgmentThis work was supported by the National Key Research and Development Programunder grant 2017YFB0802300, Applied Basic Research (Key Project) of SichuanProvince under grant 2017JY0095, and Key Foundamental Leading Project of SichuanProvince under grant 2016JY0007.References[1] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, andM. Protasi, Complexity and approximation, Springer, Berlin, 1999.150 Complexity and approximation ratio of semitotal domination in graphs[2] L. Chen, W. Zeng, and C. Lu, Np-completeness and apx-completeness of restraineddomination in graphs, Theoret. Comput. Sci. 448 (2012), 1–8.[3] M.R. Garey and D.S. Johnson, Computers and intractability: A guide to the theoryof npcompleteness (series of books in the mathematical sciences), ed, Computersand Intractability (1979), 340.[4] W. Goddard, M.A. Henning, and C.A. McPillan, Semitotal domination in graphs,Util. Math. 94 (2014), 67–81.[5] T.W. Haynes, S. Hedetniemi, and P. Slater, Fundamentals of Domination inGraphs, Marcel Dekker, New York, 1998.[6] O. [Ore, Theory of graphs, vol. 38, American Mathematical Society, 1962.[7] C.H. Papadimitriou and M. Yannakakis, Optimization, approximation, and com-plexity classes, J. Comput. System Sci. 43 (1991), no. 3, 425–440.[8] E. Zhu, Z. Shao, and J. Xu, Semitotal domination in claw-free cubic graphs, GraphsCombin. 33 (2017), no. 5, 1119–1130.

Complexity  and approximation ratio of semitotal domination in graphs

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Complexity and approximation ratio of semitotal domination in graphs

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