Discussiones Mathematicae Graph Theory 38 (2018) 385–396 doi:10.7151/dmgt.2008 THE COMPLEXITY OF SECURE DOMINATION PROBLEM IN GRAPHS Haichao Wang 1, Yancai Zhao 2 and Yunping Deng 1 1Department of Mathematics Shanghai University of Electric Power Shanghai 200090, China 2Department of Basic Science Wuxi City College of Vocational Technology Jiangsu 214153, China e-mail: [email protected] [email protected] [email protected] Abstract A dominating set of a graph G is a subset D ⊆ V (G) such that every vertex not in D is adjacent to at least one vertex in D. A dominating set S of G is called a secure dominating set if each vertex u ∈ V (G) \ S has one neighbor v in S such that (S \ {v}) ∪ {u} is a dominating set of G. The secure domination problem is to determine a minimum secure dominating set of G. In this paper, we first show that the decision version of the secure domination problem is NP-complete for star convex bipartite graphs and doubly chordal graphs. We also prove that the secure domination problem cannot be approximated within a factor of (1−ε)ln |V | for any ε> 0, unless O V NP⊆DTIME|V | (log log | |). Finally, we show that the secure domination problem is APX-complete for bounded degree graphs. Keywords: secure domination, star convex bipartite graph, doubly chordal graph, NP-complete, APX-complete. 2010 Mathematics Subject Classification: 05C69. 386 H.C. Wang, Y.C. Zhao and Y.P. Deng 1. Introduction All graphs considered in this paper are finite, simple and undirected. In a graph G with vertex set V (G) and edge set E(G), the open neighborhood of a vertex v is NG(v) = {u ∈ V (G) | uv ∈ E(G)} and the closed neighborhood is NG[v] = {v}∪ NG(v). For a subset S ⊆ V (G), the open neighborhood of S is NG(S) = Sv∈S NG(v) and the closed neighborhood of S is NG[S] = NG(S) ∪ S. An S- external private neighbor of a vertex v ∈ S is a vertex u ∈ V (G) \ S such that NG(u) ∩ S = {v}. The S-external private neighborhood of v ∈ S, denoted by epnG(v, S), is the set of all S-external private neighbors of v. The degree dG(v) of v is defined as the cardinality of NG(v). If dG(v) = 0, then v is said to be an isolated vertex. Let ∆(G) represent the maximum degree of G. In all cases above, we omit the subscript G when the graph G is clear from the context. For a subset S ⊆ V (G), the subgraph induced by S is the graph G[S] with vertex set S and edge set {uv ∈ E(G) | u,v ∈ S}. If G[S] is a complete subgraph of G, then S is called a clique of G. An independent set of G is a subset S of V (G) such that G[S] has no edge. We say that a graph G is a split graph if V (G) can be partitioned into an independent set and a clique. A graph G is said to be bipartite if the vertex set V (G) can be partitioned into two disjoint sets X and Y such that two endpoints of every edge of G lie in X and Y , respectively. We call a partition (X,Y ) of V (G) a bipartition. We write G =(X,Y,E) for a bipartite graph with bipartition (X,Y ) of V (G). A chord of a cycle is an edge joining two vertices on the cycle that are not adjacent on the cycle. A chordal graph is a graph in which every cycle of length at least four has a chord. Domination and its variations in graphs have been widely investigated as they have many applications in the real world and other disciplines such as computer networks, social networks, location theory, etc. For a detailed survey on this subject, we refer to the books [15,16] by Haynes, Hedetniemi and Slater. A subset D ⊆ V (G) is a dominating set of G if every vertex in V (G) \ D has at least one neighbor in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination problem is to find a minimum dominating set of a graph. Another variation of domination, secure domination was introduced by Cock- ayne et al. [10]. A dominating set S of G is called a secure dominating set if each vertex u ∈ V (G) \ S is adjacent to a vertex v in S such that (S \ {v}) ∪ {u} is a dominating set of G. The secure domination number γs(G) of G is the minimum cardinality of a secure dominating set of G. The secure domination problem is to find a minimum secure dominating set of a graph. An application of secure domination for graph protection is presented in [6,10]. Many of secure domination results in the literature concentrated on establishing tight bounds on γs(G) for various graph classes in terms of different graph invariants, for example, in [5,9,10,19–21]. Recently, the secure critical graphs are investigated in [7,13]. The Complexity of Secure Domination Problem in Graphs 387 On the complexity side of the secure domination problem, Merouane and Chellali [20] showed that the decision version of the secure domination problem is NP- complete for bipartite graphs and split graphs (a subclass of chordal graphs). Burger et al. [6] presented a linear time algorithm for the secure domination problem in trees. However, to the best of our knowledge, no result has been obtained on the approximability of the secure domination problem. In this paper we continue to study the complexity of the secure domination problem. Firstly, we show that the decision version of the secure domination prob- lem is NP-complete for star convex bipartite graphs and doubly chordal graphs. We also prove that the secure domination problem cannot be approximated within a factor of (1 − ε)ln |V | for any ε > 0, unless NP⊆DTIME|V |O(log log |V |). Fi- nally, we show that the secure domination problem is APX-complete for graphs with maximum degree 4. 2. Preliminaries A tree is a connected graph without cycles. A star is a special tree T = (X, F ), where X = {x0,x1,x2,...,xn} and F = {x0xi | 1 ≤ i ≤ n}. The vertex x0 is said to be the central vertex, and {x1,x2,...,xn} are called leaves. A bipartite graph G =(X,Y,E) is called star convex bipartite [17] if there exists an associated star T =(X, F ) such that NG(y) induces a subtree of T for each vertex y ∈ Y . Given a graph G =(V,E), a vertex v ∈ V is a simplicial vertex of G if NG[v] forms a clique of G. An ordering {v1,v2,...,vn} of the vertices of G is a per- fect elimination ordering (PEO) of G if vi is a simplicial vertex of the induced subgraph Gi = G[{vi,vi+1,...,vn}] for all i, 1 ≤ i ≤ n. A vertex u ∈ NG[v] is a maximum neighbor of v in G if NG[w] ⊆ NG[u] holds for each w ∈ NG[v]. A vertex v ∈ V is called doubly simplicial if it is a simplicial vertex and it has a maximum neighbor in G. An ordering {v1,v2,...,vn} of the vertices of V is a doubly perfect elimination ordering (DPEO) of G if vi is a doubly simplicial vertex of the induced subgraph Gi = G[{vi,vi+1,...,vn}] for every i, 1 ≤ i ≤ n. It is well known that a graph is chordal (doubly chordal) if and only if G has a PEO (DPEO); see [11] and [4]. To obtain our main results, we need the following result due to Cockayne et al. [10]. Proposition 1 ([10]). If S is a secure dominating set of a graph G, then the subgraph induced by epnG(v, S) is complete for each v ∈ S. 3. NP-Completeness Results of Secure Domination Problem In this section, we shall prove that the decision version of the secure domination 388 H.C. Wang, Y.C. Zhao and Y.P. Deng problem is NP-complete for star convex bipartite graphs and doubly chordal graphs. Let SDOM denote the decision version of the secure domination problem. Secure Domination Problem (SDOM) Instance: A graph G =(V,E) and a positive integer k ≤|V |. Question: Does G have a secure dominating set of cardinality at most k? To show that SDOM is NP-complete for star convex bipartite graphs, we will make use of the well-known domination problem (DOM) which is NP-complete for bipartite graphs [3]. Domination Problem (DOM) Instance: A graph G =(V,E) and a positive integer k ≤|V |. Question: Does G have a dominating set of cardinality at most k? Theorem 2. The SDOM problem is NP-complete for star convex bipartite graphs. Proof. The SDOM problem is clearly in NP. We now describe a polynomial transformation from the DOM problem for bipartite graphs to the SDOM prob- lem for star convex bipartite graphs. Given a bipartite graph G = (X,Y,E) with X = {x1,x2,...,xs} and Y = ′ ′ ′ ′ ′ {y1,y2,...,yt}, we construct the graph G =(X ,Y ,E ) as follows. Let X = X∪ ′ ′ {x0,x}, Y = Y ∪{y0,y}, and E = E∪{xyi | 0 ≤ i ≤ t}∪{yxi | 0 ≤ i ≤ s}∪{xy}. Note that the graph G′ is a star convex bipartite graph with an associated star ′ ′ T =(X , F ), where F = {xxi | 0 ≤ i ≤ s}.