On minimum secure dominating sets of graphs

Transcription

1 On minimum secure dominating sets of graphs AP Burger, AP de Villiers & JH van Vuuren December 24, 2013 Abstract A subset X of the verte set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each verte u not in X, there is a neighbouring verte v of u in X such that the swap set (X\{v}) {u} is again a dominating set of G, in which case v is called a defender. The secure domination number of G is the cardinalit of a smallest secure dominating set of G. In this paper, we show that ever graph of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3. 1 Introduction Let G = (V, E) be a simple graph of order n. The (open) neighbourhood of a verte u V is the set of all vertices that are adjacent to u in G and is denoted b N(u). A set D V is a dominating set of G if each verte u V \D is adjacent to a verte in D. A dominating set X V of G is secure dominating if, for each u V \X, there eists a verte v N(u) X such that the swap set (X\{v}) {u} is again a dominating set of G, in which case v is called a defender and is said to defend u. A secure dominating set of G of minimum cardinalit is called a minimum secure dominating set of G and this minimum cardinalit is called the secure domination number of G, denoted b γ s (G). The notion of secure domination has been researched etensivel. A number of general bounds have, for eample, been established for the parameter γ s (G) in [6]. Eact values of γ s (G) were also established in [6] for various graph classes, such as paths, ccles, complete multipartite graphs and products of paths and ccles. Furthermore, various aspects of secure domination and properties of the secure domination number of a graph have been researched in [1, 2, 3, 4, 5, 9]. This paper opens, in 2, with some basic definitions and a generic description of the structure of an secure dominating set of a graph, after which we characterise, in 3, those classes of graphs G for which γ s (G) = 1, 2 or 3. We finall show in 4 that ever graph Department of Logistics, Stellenbosch Universit, Private Bag X1, Matieland, 7602, South Africa, fa: , s: apburger@sun.ac.a and antondev@sun.ac.a Department of Industrial Engineering, Stellenbosch Universit, Private Bag X1, Matieland, 7602, South Africa, fa: , vuuren@sun.ac.a 1

2 2 The structure of a secure dominating set 2 of minimum degree at least 2 possesses a minimum secure dominating set in which all vertices are defenders. 2 The structure of a secure dominating set Let X be an subset of the verte set V of a simple graph G = (V, E) and let v X. Then a verte V \X is an eternal private neighbour of v with respect to X if N() X = {v}. We denote the set of all eternal private neighbours of v with respect to X b Epn G (v, X). The verte set V ma be partitioned into five subsets with respect to X. Denote b X P the set of vertices in X which have private neighbours eternal to X, and let P X be the set of all these eternal private neighbours. Furthermore, let X D = X\X P and let D X be the set of vertices in V \(X P X ) that are dominated b X D. Finall, let U X be the set of vertices not in X, P X or D X. Then, as illustrated in Figure 2.1. V = X P X }{{ D P } X U X D X, (2.1) X X P X D X v 1 v 2 v 3 v 4 v 5 v 6 v 7 V X u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u 12 u 13 u 14 u 15 P X U X D X Figure 2.1: An eample of the verte set partition in (2.1). It follows that each verte in D X is defended b at least one verte in X D. Whilst ever verte in P X is further dominated b eactl one verte in X P, some of the vertices in P X ma not be defended b an vertices in X P. Even worse, some vertices in U X ma not even be dominated b an vertices in X. For eample, in Figure 2.1 the vertices u 1, u 2 P X are not defended (b v 1 ), because G[{v 1, u 1, u 2 }] is not complete. The vertices u 3, u 4 P X are, however, defended (b v 2 ), because G[{v 2, u 3, u 4 }] is complete. The vertices u 5, u 6, u 7 P X are similarl defended (b v 3, v 4 and v 4, respectivel). Furthermore, the verte u 8 U X is defended (b v 4 ) since G[{v 4, u 6, u 7, u 8 }] is complete. However, u 9 U X is defended b neither v 3 nor v 4, because both G[{v 3, u 5, u 9 }] and G[{v 4, u 6, u 7, u 9 }] are incomplete. Finall, u 10, u 11, u 12 U X are not even dominated b X, let alone defended.

3 3 Graphs with small secure domination numbers 3 The set X is a secure dominating set of G if and onl if the following three properties are satisfied [6, Proposition 2]: (1) Ever verte in U X is dominated b at least one verte in X P, ( The private neighbours in P X of each verte in X P form a clique in G, (3) For each verte u U X there is a verte v X P such that G[Epn G (v, X) {u, v}] is complete. Note that there are no edges between vertices in X D and vertices in U X. 3 Graphs with small secure domination numbers In this section, we characterise those classes of graphs that have secure domination numbers 1, 2 or 3. We start with the simplest case graphs with secure domination number 1. Proposition 1 (Characterisation of graphs with secure domination number 1) A graph has secure domination number 1 if and onl if it is a complete graph. Proof: Clearl, γ s (K n ) = 1 [6, Proposition 9]. Conversel, suppose G is a graph of order n 3 that is not complete, but suppose, to the contrar, that γ s (G) = 1 and let X = {} be a dominating set of G for some V (G). Since G contains two non-adjacent vertices u and v, it follows that / {u, v}, for otherwise X would not be dominating. But then neither u nor v is defended b, a contradiction. Therefore, γ s (G) 2. In order to characterise graphs with secure domination number 2, the following graph construction is required. Let i, j be positive integers and let k, l be non-negative integers. Let Φ(i, j, k, l) denote the graph of order i j k l and sie ( ( i j k(i 1) l(j 1) containing two verte-disjoint cliques K i and K j of orders i and j, respectivel, together with two further disjoint verte subsets U k and U l of vertices of cardinalities k and l, respectivel, to which the following edges are added: (1) Each verte in K i is joined to all vertices of U k (if U k ). ( Each verte in K j is joined to all vertices of U l (if U l ). (3) Some verte V (K i ) is joined to all vertices in U l, and some verte V (K j ) is joined to all vertices in U k. Note that no two vertices of U k are adjacent, and similarl for U l. The construction of Φ(i, j, k, l) is illustrated in Figure 3.1. Proposition 2 (Characterisation of graphs with secure domination number A graph has secure domination number 2 if and onl if it is not complete and contains Φ(i, j, k, l) in Figure 3.1 as spanning subgraph for some integers i, j 1 and k, l 0. Proof: The set {, } is clearl a secure dominating set of cardinalit 2 for Φ(i, j, k, l). Since γ s (Φ(i, j, k, l)) 1 b Proposition 1, it follows that γ s (Φ(i, j, k, l)) = 2, which settles the sufficienc.

4 3 Graphs with small secure domination numbers 4 U k K i K j U l Figure 3.1: The graph Φ(i, j, k, l). For the necessit, suppose X = {, } is a secure dominating set of some graph G of order n and consider the partition in (2.1). There are eactl three cases to consider in terms of the possible structure of G. Case 1:, X P. In this case both X D and D X in Figure 2.1 are empt and the vertices in U X are defended b vertices in X P. If defends some verte u U X, then G[Epn G (, X P ) {, u}] forms a clique in G b [6, Proposition 2]. Similarl, if defends some verte v U X, then G[Epn G (, X P ) {, v}] forms a clique in G, as depicted in Figure 3.2. Let U k be the set of k 0 vertices that form a clique, K i (sa), together with and its private neighbours, and let U l be the set of l 0 vertices that form a clique, K j (sa), together with and its private neighbours. Note that U k and/or U l ma possibl be empt. In this case the graph G therefore contains the graph Φ(i, j, k, l) in Figure 3.1 as spanning subgraph (where i j k l = n). X P P X v u U X Figure 3.2: A spanning subgraph of G in Case 1 of the proof of Proposition 2. Case 2: X P and X D. In this case U X = in Figure 2.1 and each verte in D X is adjacent to both X P and X D, as depicted in Figure 3.3. Furthermore, the private neighbours of form a clique, K i (sa), in G together with b [6, Proposition 2]. In this case, therefore, U k =, K j contains onl the verte, and U l contains all the vertices in D X in Figure 3.1. Hence G contains the graph Φ(i, 1, 0, l) in Figure 3.1 as spanning subgraph (where i l 1 = n).

5 3 Graphs with small secure domination numbers 5 X P X D P X D X Figure 3.3: A spanning subgraph of G in Case 2 of the proof of Proposition 2. Case 3:, X D. In this case both P X and U X in Figure 2.1 are empt, as depicted in Figure 3.4. Therefore, K i contains onl the verte, K j contains onl the verte, and D X = U k U l in Figure 3.1 (where 2 k l = n). Therefore G contains the graph Φ(1, 1, k, l) in Figure 3.1 as spanning subgraph. X D D X Figure 3.4: A spanning subgraph of G in Case 3 of the proof of Proposition 2. The characterisation in Proposition 2 ma be used to prove succinctl that γ s (G) = 2 for an incomplete graph that contains Φ(i, j, k, l) as spanning subgraph, b merel citing the parameters of the spanning subgraph Φ(i, j, k, l) as certificate. Note that multiple certificates ma eist showing that γ s (G) = 2 for a graph or graph class G. Consider, as an eample, the connected dumbbell graph D a,b of order a b and sie ( a ( b ) 2 1 obtained b joining two verte disjoint cliques of orders a and b b means a single edge. A graphical presentation of the dumbbell graph D 3,4 is shown in Figure 3.5(a), while it is illustrated in Figure 3.5(b) that Φ(2, 4, 1, 0) is a spanning subgraph of D 3,4. K 2 U 1 K 4 (a) D 3,4 (b) Φ(2, 4, 1, 0) D 3,4 Figure 3.5: The dumbbell graph D 3,4. Since the Φ(a 1, b, 1, 0) is a spanning subgraph of the dumbbell graph D a,b in general, it follows immediatel from Propositions 1 and 2 that γ s (D a,b ) = 2 for all a, b N 0. Further eamples of certificates showing that γ s (G) = 2 for various infinite classes of graphs G are shown in Table 3.1. Figure 3.6 shows four of the certificates listed in Table 3.1 in the contet of the full graph classes in the table.

6 3 Graphs with small secure domination numbers 6 Graph class Certificate The complete graph less an edge, K n e Φ(n 1, 1, 0, 0) The complete graph less two edges, K n 2e Φ(n 2, 2, 0, 0) The complete bipartite graph, K 2,n 2 Φ(2, 1, 0, n 3) The book graph B n = K 2 K n 2 Φ(2, 1, 0, n 3) The cartesian product, K 2 K a Φ(a, a, 0, 0) The dumbbell graph, D a,b Φ(a 1, b, 1, 0) Table 3.1: Certificates showing that γ s (G) = 2 for various infinite classes of graphs G. K 2 K n 1 (a) Φ(n 1, 1, 0, 0) K n e U n 3 (b) Φ(2, 1, 0, n 3) K 2,n 2 K a K a K a U 1 K a (c) Φ(a, a, 0, 0) K 2 K a (d) Φ(a 1, b, 1, 0) D a,b Figure 3.6: Various graph classes with certificates. Dotted lines represent edges that are present in the graph classes of Table 3.1, but not in the certificates listed in the table. (a) The complete graph less an edge. (b) The complete bipartite graph. (c) The cartesian product K 2 K a. (d) The dumbbell graph. We close this section b characterising graphs G for which γ s (G) = 3. The following graph constructions are required for this characterisation. Let i, j, k be positive integers and let r, s, t be non-negative integers. Let Ψ(i, j, k, r, s, t) denote the graph of order i j k r s t and sie ( ( i j ( k r(i 1) s(j 1) t(k 1) containing three verte-disjoint cliques K i, K j and K k of orders i, j and k, respectivel, together with three further disjoint verte subsets U r, U s and U t of cardinalities r, s and t, respectivel, to which the following edges are added: (1) Each verte in U r is joined to all vertices in K i. ( Each verte in U s is joined to all vertices in K j.

7 3 Graphs with small secure domination numbers 7 (3) Each verte in U t is joined to all vertices in K k. (4) Vertices V (K i ), V (K j ) and V (K k ) are identified, joining (a) each verte in U r to either or (but not both). (b) each verte in U s to either or (but not both). (c) each verte in U t to either or (but not both). Note that no two vertices in U r are adjacent, and similarl for U s and U t. The construction of the graph Ψ(i, j, k, r, s, t) is illustrated in Figure 3.7(a). K i K j U r U s U r U s U t K i K j U t K k (a) Ψ(i, j, k, r, s, t) (b) Ψ (i, j, r, s, t) Figure 3.7: The graphs Ψ(i, j, k, r, s, t) and Ψ (i, j, r, s, t). Another graph, denoted b Ψ (i, j, r, s, t), is required for the characterisation of graphs G for which γ s (G) = 3, where i and j are positive integers and r, s and t are non-negative integers. This graph has order 1ij r st and sie ( ( i j r(i1)s(j 1)2t, and contains three verte-disjoint cliques K i, K j and of orders i, j and 1, respectivel, together with three further disjoint verte subsets U r, U s and U t of cardinalities r, s and t, respectivel, to which the following edges are added: (1) Each verte in U r is joined to all vertices in K i. ( Each verte in U s is joined to all vertices in K j. (3) Let be the verte of and identif two vertices V (K i ) and V (K j ), joining (a) each verte in U r to. (b) each verte in U s to. (c) each verte in U t to and to either or (but not both). Note again that no two vertices in U r are adjacent, and similarl for U s and U t. construction of the graph Ψ (i, j, r, s, t) is illustrated in Figure 3.7(b). The Proposition 3 (Characterisation of graphs with secure domination number 3) An incomplete graph has secure domination number 3 if and onl if it does not contain the graph Φ(i, j, k, l) in Figure 3.1 as subgraph for an integers i, j 1 and k, l 0, but contains one of the graphs Ψ(i, j, k, r, s, t) or Ψ (i, j, r, s, t) in Figure 3.7 as spanning subgraph for some integers i, j, k 1 and r, s, t 0.

8 3 Graphs with small secure domination numbers 8 Proof: The set {,, } is clearl a secure dominating set of cardinalit 3 for both the graphs Ψ(i, j, k, r, s, t) and Ψ (i, j, r, s, t). Furthermore, γ s (Ψ(i, j, k, r, s, t)) 1, 2 and γ s (Ψ (i, j, r, s, t)) 1, 2 b Propositions 1 and 2. It follows that γ s (Ψ(i, j, k, r, s, t)) = γ s (Ψ (i, j, r, s, t)) = 3, which settles the sufficienc. For the necessit, suppose X = {,, } is a secure dominating set of some graph G of order n and again consider the partition in (2.1). There are eactl four cases to consider in terms of the possible structure of G. X P P X U X Figure 3.8: A spanning subgraph of G in Case 1 of the proof of Proposition 3. Case 1:,, X P. In this case both X D and D X in Figure 2.1 are empt and the vertices in U X are defended b vertices in X P. Therefore each verte in U X is adjacent to at least two vertices in X P, as depicted in Figure 3.8. Let U r be the set of r 0 vertices that form a clique, K i (sa), together with and its private neighbours according to [6, Proposition 2]. Similarl, let U s be the set of s 0 vertices that form a clique, K j (sa), together with and its private neighbours, and let U t be the set of t 0 vertices that form a clique, K k (sa), together with and its private neighbours. Note that U r, U s and/or U t ma possibl be empt. Since the vertices in U r are defended b and each verte in U X is defended b at least two vertices in X P, it follows that all the vertices in U r are adjacent to or. Similarl, all the vertices in U s are adjacent to or, and all the vertices in U t are adjacent to or. In this case G therefore contains the graph Ψ(i, j, k, r, s, t) in Figure 3.7(a) as spanning subgraph (where i j k r s t = n). X P X D P X U X D X Figure 3.9: A spanning subgraph of G in Case 2 of the proof of Proposition 3. Case 2:, X P and X D. In this case each verte in D X is adjacent to at least one verte of X P and each verte in U X is adjacent to both vertices of X P in Figure 2.1, as depicted in Figure 3.9. Let U r be the set of r 0 vertices that form a clique, K i (sa), together with and its private neighbours according to [6, Proposition 2]. Similarl, let U s be the set of s > 0 vertices that form a clique, K j (sa), together with and its private neighbours. For t 0, let each verte in D X = U t be adjacent to, where K k contains onl the verte. Then each verte of U r is adjacent to, each verte of U s is adjacent to, each verte of U t is adjacent to and to either or. In this case G therefore contains

9 3 Graphs with small secure domination numbers 9 the graph Ψ (i, j, r, s, t) in Figure 3.7(b) as spanning subgraph (where ij r st = n). X P X D P X D X Figure 3.10: A spanning subgraph of G in Case 3 of the proof of Proposition 3. Case 3: X P and, X D. In this case U X is empt in Figure 2.1, for otherwise an verte u U X would be a private neighbour of, and each verte in D X is furthermore adjacent to at least one verte in X D, as depicted in Figure Therefore, K j contains onl the verte, K k contains onl the verte, D X = U s U t and U r = in Figure 3.7(a) and so G contains the graph Ψ(i, 1, 1, 0, s, t) in Figure 3.7(a) as spanning subgraph (where i s t 2 = n). X D D X Figure 3.11: A spanning subgraph of G in Case 4 of the proof of Proposition 3. Case 4:,, X D. In this case X P, P X and U X are all empt in Figure 2.1 and each verte in D X is adjacent to at least two vertices in X D, as depicted in Figure Therefore, K i contains onl the verte, K j contains onl the verte, K k contains onl the verte, and D X = U r U s U t in Figure 3.7(a) and so G contains the graph Ψ(1, 1, 1, r, s, t) in Figure 3.7(a) as spanning subgraph (where i s t 3 = n). The characterisation in Proposition 3 ma be used to prove succinctl that γ s (G) 3 for a graph that contains either Ψ(i, j, k, r, s, t) or Ψ (i, j, r, s, t) in Figure 3.7 as spanning subgraph, b merel citing the particular spanning subgraph as certificate. Showing that γ s (G) 2 for such graphs ma, however, prove more cumbersome. Figure 3.12: The double dumbbell graph D 3,5,4. Consider, as an eample, the connected graph of order a b c and sie ( ( a b ( c 2 obtained b joining three verte disjoint cliques of orders a, b and c in a chain b means of two additional edges, as illustrated in Figure 3.12(a). We call this graph a double dumbbell graph and denote it b D a,b,c. Since the Ψ(a 1, b 1, c, 1, 1, 0) is a spanning

10 4 On the defenders in a minimum secure dominating set 10 subgraph of the double dumbbell graph D a,b,c, it follows immediatel from Proposition 3 that γ s (D a,b,c ) 3. Further eamples of certificates showing that γ s (G) 3 for various infinite classes of graphs G are shown in Table 3.2. Figure 3.13 shows four of the certificates listed in Table 3.2 in the contet of the full graph classes in the table. Graph class Certificate The complete bipartite graph less two edges, K 2,n 2 2e Ψ(2, 1, 1, 0, n 4, 0) The complete tripartite graph,a,b with 3 a b Ψ (1, 1, 0, 0, a b The complete tripartite graph K a,b,c with 3 a b c Ψ(1, 1, 1, b 1, c 1, a 1) The cartesian product, K 3 K s Ψ(s, s, s, 0, 0, 0) The double dumbbell graph, D a,b,c Ψ(a 1, b 1, c, 1, 1, 0) Table 3.2: Certificates showing that γ s (G) 3 for various infinite classes of graphs G. 4 On the defenders in a minimum secure dominating set It is interesting to note that all members of a minimum secure dominating set need not be defenders, as illustrated in Figure 4.1. We show that it is possible to increase the number of defenders successivel in a minimum secure dominating set of an connected graph without end vertices, until all the members of the set are defenders. Proposition 4 If X is a minimum secure dominating set of a connected graph G with minimum degree at least 2 and some verte in X is not a defender, then there eists another minimum secure dominating set of G which contains one more defender than X. Proof: Let G be a connected graph with minimum degree at least 2 and let X be a minimum secure dominating set of G in which some verte X does not defend an verte in N(). Then Epn G (, X) =, for otherwise Epn G (, X) would induce a clique in G, all of whose vertices are defended b. We first show, b contradiction, that N() X. Suppose, to the contrar, that N() (V \X) and let v be a neighbour of in V \X. Then v is adjacent to some verte in X\{}, and so the swap set (X\{}) {v} is a dominating set of G. Therefore, defends v, a contradiction. We show net, again b contradiction, that Epn G (u, X) for all u N(). Suppose, to the contrar, that Epn G (u, X) = for some verte u N(). Then each verte in N(u ) (V \X) is dominated b at least one verte in X\{, u } and u defends all vertices in N(u ) (V \X). But then X\{} is a secure dominating set of G, since each verte in V \X is defended b a verte in X while is defended b u, contradicting the minimalit of X. Now let w N() and suppose w Epn(w, X), as illustrated in Figure 4.2. Since the minimum degree of G is at least two, w is adjacent to some verte w V (G)\X. Let D X be the set of all vertices that defend w. Then X = (X\{}) {w } is a secure

11 4 On the defenders in a minimum secure dominating set 11 U c 1 U b 1 b 1 a 1 }{{}}{{} U a 1 U ab 2 (a) Ψ (1, 1, 0, 0, a b,a,b (b) Ψ(1, 1, 1, b 1, c 1, a 1) K a,b,c K s K s K a 1 U 1 U 1 K b 1 K c K s (c) Ψ(s, s, s, 0, 0, 0) K 3 K s (d) Ψ(a 1, b 1, c, 1, 1, 0) D a,b,c Figure 3.13: Various graph classes with certificates. Dotted lines represent edges that are present in the graph classes of Table 3.2, but not in the certificates listed in the table. (a) The complete multipartite graph,a,b with 3 a b. (b) The complete multipartite graph K a,b,c with 3 a b c. (c) The cartesian product K 3 K a. (d) The double dumbbell graph. dominating set of G in which w defends, w defends all the vertices in N(w ) (V \X ), w is defended b D {w } and the remaining vertices of V \X are defended b the vertices that defended them in X. The repeated application of Proposition 4 is illustrated in Figure 4.3. Although there are graphs with minimum degree 1 which also admit minimum secure dominating sets in which all vertices are defenders, such as the star,n 1, the result of Corollar 1 does not necessaril hold for graphs with minimum degree 1 the path P 3

12 4 On the defenders in a minimum secure dominating set 12 v 1 v 3 v 5 v 2 v 4 v 6 Figure 4.1: A graph G 1 of order 6 with a minimum secure dominating set denoted b solid vertices; v 3 defends v 5, while v 4 defends v 6. Although members of the minimum secure dominating set, the vertices v 1 and v 2 are not defenders. X N() w D w w Figure 4.2: The secure dominating set in the proof of Proposition 4. of order 3 is the smallest counter eample. The following result is an immediate consequence of Proposition 4. Corollar 1 If G is a connected graph with minimum degree at least 2, then there eists a minimum secure dominating set of G in which ever verte is a defender. (a) (b) (c) Figure 4.3: Minimum secure dominating sets of a graph G 2 of order 8 with minimum degree 2, containing (a) two, (b) three and (c) four defenders. Acknowledgements Research towards this paper was financiall supported b the South African National Research Foundation (GUNs 70593, and 81558). The last author was additionall supported b a grant from Subcommittee A of Stellenbosch Universit s Research Committee.

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