p-liar s Domination in a Graph

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1 Applied Mathematical Sciences, Vol 9, 015, no 107, HIKARI Ltd, wwwm-hikaricom p-liar s Domination in a Graph Carlito B Balandra 1 Department of Arts and Sciences Southern Philippines Agri-Business Marine and Aquatic School of Technology Malita, Davao del Sur, 800, Philippines Sergio R Canoy, Jr, and Imelda S Aniversario Department of Mathematics and Statistics College of Science and Mathematics MSU-Iligan Institute of Technology Tibanga, Iligan City, Philippines Copyright c 015 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract A set S V (G) is a p-liar s dominating set (plds) of graph G if (i) N G (v) S for every v V (G) \ S (that is, S is a - dominating set of G), and (ii) [N G (u) N G (v)] S 3 for any two distinct vertices u, v V (G) \ S The p-liar s domination number of G, denoted by γ plr (G), is the smallest cardinality of a p-liar s dominating set of G In this paper, we study the concept of p-liar s domination of a graph G and investigate it for graphs resulting from the binary operations join and corona 1 Introduction Let G = (V (G), E(G)) be a simple connected graph The neighborhood of v V (G) is the set N G (v) = {x V (G) : xv E(G)} A vertex 1 Research is funded by the Commission on Higher Education (CHED), Philippines under Faculty Development Program Phase II

2 533 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario v in a graph G is isolated if deg G v = 0; it is an end-vertex or leaf if deg G v = 1 Denote by I(G) and L(G) the set of all isolated vertices and leaves of G, respectively Let S V (G) of a graph G A vertex w is an external private neighbor (abbreviated epn) of v S if w V (G) \ S and N(w) S = {v} The set of all external private neighbors of v is denoted by epn(v; S) A set S V (G) is a dominating set of G if for every v V (G) \ S, there exists u S such that uv E(G), that is, N G [S] = V (G) The domination number of G, denoted by γ(g), is the smallest cardinality of a dominating set of G Any dominating set in G of cardinality γ(g) is referred to as a γ-set of G A subset S of V (G) is called an almost dominating set of G if V (G) \ N[S] 1 The a-domination number of G, denoted by γ a (G), is the smallest cardinality of an almost dominating set of G An almost dominating set of G with cardinality γ a (G) is referred to as a γ a -set of G A set S V (G) is a liar s dominating set (lds) of G if (i) N G [v] S for every v V (G) (that is, S is a double dominating set of G), and (ii) (N G [u] N G [v]) S 3 for any two distinct vertices u, v V (G) A set S V (G) is a p-liar s dominating set (plds) of G if (i) N G (v) S for every v V (G) \ S (that is, S is a - dominating set of G), and (ii) [N G (u) N G (v)] S 3 for any two distinct vertices u, v V (G) \ S The p-liar s domination number of G, denoted by γ plr (G), is the smallest cardinality of a p-liar s dominating set of G Any subset of V (G) with cardinality γ plr (G) is called a γ plr -set of G Domination in graph as well some of its variations can be found in [1] The concept of liar s domination of a graph G was introduced by Slater and Roden in [3] About five years later, Sterling [4] studied the concept for grid graphs A similar study of the concept is also considered in [] Results It is easy to see that the vertex set of a graph is p-liar s dominating set Hence the following observation is immediate Remark 1 Let G be a graph of order n Then 1 γ plr (G) n Theorem Let G be a graph Then (i) γ plr (G) = 1 if and only if G = K 1 ; and (ii) γ plr (G) = if and only if G {K, K, K 3, P 3 } Proof :

3 p-liar s domination, join and corona of graphs 5333 (i) Suppose γ plr (G) = 1 and let S = {x} be a plds of G If there exists y V (G) \ {x}, then xy E(G) and N G (x) S = 1 Hence, S is not plds of G, contrary to our assumption Thus, V (G) = {x}, that is, G = K 1 The converse is clear (ii) Let S = {x, y} be a γ plr -set of G Then by Definition and Theorem 3, V (G) = or V (G) = 3 If V (G) =, then G = K or G = K If V (G) = 3, then G = P 3 or G = K 3 The converse is straightforward Remark 3 Let G be a connected graph of order n 4 Then γ plr (G) 3 Remark 4 Let G be a graph dominating set S of G Then I(G) L(G) S for every liar s Theorem 5 Let n be a positive integer Then 1, if n = 1 (i) γ plr (K n ) =, if n = or n = 3 3, if n 4; (ii) γ plr (P n ) = n+1 for n 1; and 3, if n = 4 n+1 (iii) γ plr (C n ) =, if n is odd, if n is even (n 4) Proof : n (i) Let G = K n If n = 1, then γ plr (G) = 1 by Theorem If n = or 3, then γ plr (G) = by Theorem 3 Suppose n 4 Then γ plr (G) 3 by Remark 4 Pick any distinct vertices x, y, z of G and let S = {x, y, z} Then S is a p-liar s dominating set of G Thus, γ plr (G) = 3 (ii) Clearly, γ plr (P 1 ) = 1, and γ plr (P ) = γ plr (P 3 ) = = 3 = 4 by Theorem and Theorem 3, respectively Let n 4 and let P n = [x 1, x,, x n 1, x n ] Let S be a γ plr -set of G Then x 1, x n S by Remark 4 Also, x / S or x n 1 / S Assume that x / S and consider the following cases: Case 1: n = k for some positive integer k Then S = {x 1, x 3,, x (k 1)+1, x n } Hence, γ plr (G) = S = k + 1 = n + 1 = n +

4 5334 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario Case : n = k + 1 for some positive integer k Then S = {x 1, x 3,, x (k 1)+1, x n } Hence, γ plr (G) = S = k + 1 = n 1 Accordingly, γ plr (G) = n = n + 1 (iii) Clearly, γ plr (C 3 ) = 3+1 = and γ plr (C 4 ) = 3 by Theorem 33 Suppose n 5 and let C n = [x 1, x,, x n, x 1 ] Let S be γ plr -set of C n and assume that x 1 S Then x n S or x n 1 S Assume x n S and consider the following cases: Case 1: n = k for some positive integer k Then S = {x 1, x 3,, x (k 1) 1, x (k 1)+1 } Thus, S = k = n Since x n S, S = k + 1 = n + 1 = n+ Since n < n+, it follows that γ plr (G) = n Case : n = k + 1 for some positive integer k Then S = {x 1, x 3,, x (k 1), x n } Hence, γ plr (G) = S = k + 1 = n = n+1 This completes the proof of the theorem Theorem 6 Let G be a graph of order n 3 Then γ plr (G) = n if and only if for each component C of G, C = K 1 or C = K Proof : Suppose γ plr (G) = n Let C be a component of G Suppose that C is neither K 1 nor K Then V (C) 3 and there exists v V (C) such that deg G v Let S = V (G) \ {v} Then S is p-liar s dominating set of G Thus γ plr (G) S = n 1, contrary to our assumption Therefore, C = K 1 or C = K The converse follows from Remark 4 Theorem 7 Let G be a graph of order n 4 Then γ plr (G) = 3 if and only if there exists S V (G) with S = 3 such that S is a -dominating set and for each pair of distinct vertices u, v V (G)\S, either N G (u) S N G (v) S or N G (u) S = N G (v) S = S Proof : Suppose γ plr (G) = 3 Let S be γ plr -set of G Then S = 3 and by Definition of p-liar s (i), S is a -dominating set of G Let u, v V (G) \ S with u v Suppose N G (u) S = N G (v) S Since S is p-liar s dominating set, (N G (u) S) (N G (v) S) = (N G (u) N G (v)) S = N G (u) S = 3 by Definition p-liar s (ii) This implies that N G (u) S = N G (v) S = S For the converse, suppose there exists S V (G) satisfying the given conditions Then S is a p-liar s dominating set of G Since n 4, S is a γ plr -set of (G) by Remark 3 Thus, γ plr (G) = S = 3

5 p-liar s domination, join and corona of graphs 5335 Corollary 8 Let G be a connected graph of order n = 4 Then γ plr (G) = 3 3 p-liar s Domination in the Join of Graphs The join of two graphs G and H is the graph G+H with vertex-set V (G+H) = V (G) V (H) and edge-set E(G + H) = E(G) E(H) {uv : u V (G), v V (H)} Theorem 31 Let G be a graph of order n and K 1 = {v} Then S V (K 1 + G) is a p-liar s dominating set of K 1 + G if and only if one of the following holds: (i) S is a p-liar s dominating set of G, where S = V (G) whenever n = or 3 (ii) S = S 1 {v}, where S 1 is a dominating set of G and epn(x; S 1 ) 1 for all x S 1 Proof : Suppose S is plds of K 1 + G Consider the following cases: Case 1 v / S Then S V (G) Since S is a p-liars dominating set of K 1 + G, S is a plds of G If n =, then K 1 + G is P 3 or K 3 It follows from Theorem 3 that S = V (G) If n = 3, then γ plr (K 1 + G) = 3 by Corollary 8 Hence, S = V (G) Case v S Then S = S 1 {v}, where S 1 V (G) Let z V (G) \ S 1 Since S is a -dominating set of K 1 + G and v S N K1 +G(z), it follows that zy E(G) for some y S 1 Hence, S 1 is a dominating set of G Now, let x S 1 and suppose that epn(x; S 1 ) Let a, b epn(x; S 1 ), where a b Then N G (a) S 1 = N G (b) S 1 = {x} Thus, N K1 +G(a) S = N K1 +G(b) S = {x, v} This implies that S does not satisfy (ii) of Definition of p-liar s, contradicting our assumption that S is a plds of K 1 + G Therefore, epn(x; S 1 ) 1 For the converse, suppose that (i) holds Then, clearly, S is a plds of K 1 + G Next, suppose that (ii) holds Let u V (K 1 + G) \ S Then u V (G) \ S 1 Since S 1 is a dominating set of G, N G (u) S 1 1 Hence, N K1 +G(u) S = (N G (u) S 1 ) {v} This shows that S is a -dominating set of K 1 +G Now, let z, w V (K 1 +G)\S Then z, w V (G)\S 1 If none of z and w is an external private neighbor of any element of S 1, then N G (z) S 1 and N G (w) S 1 Hence, (N K1 +G(z) S) (N K1 +G(w) S) 3 Suppose now that one of z and w, say z, is an external private neighbor of x S 1 Then, by assumption, w / epn(x; S) This implies that N G (z) S 1 = {x} = N G (w) S 1

6 5336 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario Hence, N K1 +G(z) S = {x, v} = N K1 +G(w) S, showing that (N K1 +G(z) N K1 +G(w)) S 3 Therefore, S is a plds of K 1 + G As a consequence of Theorem 31, we have the Corollary 3 Corollary 3 Let G be a graph of order n 4 Then γ plr (K 1 + G) = min{γ plr (G), γ (G) + 1}, where γ (G) = min { D : D is a dominating set of G with epn(x; D) 1 for all x D} Proof : Let S be a γ P LR -set and D be a γ -set of G Then S and S = D {v} are plds of K 1 + G, by Theorem 31 Thus, γ plr (K 1 + G) min{ S, S } = min{γ plr (G), γ (G) + 1} Next, let S be a γ plr -set of K 1 + G By Theorem 31, S is a plds of G or S = S 1 {v}, where S 1 is a dominating set of G with epn(x; S 1 ) 1 for all x S 1 Hence, γ plr (K 1 + G) = S min{γ plr (G), γ (G) + 1} This proves the desired equality Example 33 Consider the graphs K 1 + K 5 and K 1 + G in Figure 1 K 1 + K 5 K 1 + G Figure 1: The graphs K 1 + K 5 and K 1 + G Now, γ plr (K 5 ) = 5 = γ (K 5 ) Then γ plr (K 5 ) = 5 < = 1 + γ (K 5 ) Hence, γ plr (K 1 + G) = 5 = γ plr (K n ) Also, γ plr (G) = 4 and γ (G) = Thus 1 + γ (G) = 1 + = 3 < 4 = γ plr (G) Hence, γ plr (K 1 + G) = 3 = 1 + γ (G) Theorem 34 Let G and H be non-trivial connected graphs Then S V (G + H) is a p-liar s dominating set of G + H if and only if one of the following holds: (i) S is a p-liar s dominating set of G

7 p-liar s domination, join and corona of graphs 5337 (ii) S is a p-liar s dominating set of H (iii) S V (G) 3 and S V (H) 3 (iv) S = S 1 S, where S 1 V (G) and S V (H) and satisfy the following: (a) S 1 is a dominating set of G such that S 1 and epn(x; S 1 ) 1 for all x S 1, and (b) S = 1, where V (H) \ N H [S ] 1 whenever S 1 = (v) S = S 1 S, where S 1 V (H) and S V (G) and satisfy the following: (a) S 1 is a dominating set of H such that S 1 and epn(x; S 1 ) 1, for all x S 1, and (b) S = 1, where V (G) \ N G [S ] 1 whenever S 1 = (vi) S = S 1 S, where S 1 V (G) and S V (H) and satisfy the following: (a) S 1 = and V (G) \ N G [S 1 ] 1, and (b) S = and V (H) \ N H [S ] 1 (vii) S = S 1 S, where S 1 V (G) and S 1 = and S V (H), S 3 and V (H) \ N H [S ] 1 (viii) S = S 1 S, where S 1 V (H) and S 1 = and S V (G), S 3 and V (G) \ N G [S ] 1 Proof : Suppose S is a plds of G + H If S V (H) = or S V (G) =, then S is a plds of G or H Thus, (i) or (ii) holds Now, suppose that S 1 = S V (G) and S = S V (H) If S 1 3 and S 3, then (iii) holds Consider the following cases: Case 1 S 1 and S = 1 or S 1 = 1 and S Suppose that S 1 and S = 1 Let x V (G) \ S 1 Since S is a -dominating set of G + H, N G+H (x) S = S + N G (x) S 1 = 1 + N G (x) S 1 This implies that N G (x) S 1 1 Hence, S 1 is a dominating set of G Suppose that epn(x; S 1 ) Let y, z be distinct external private neighbors of x in S 1 Then N G+H (y) S = N G+H (z) S = {x} S This implies that S does not satisfy Definition of p-liar s (ii), contrary to our assumption that S is p-liar s dominating set of G + H Therefore, epn(x; S 1 ) 1 for all x S 1 Next, suppose that S 1 = Suppose further that V (H) \ N H [S ] Then there exist distinct vertices a, b V (H) \ N H [S ] such that [N G+H (a) N G+H (b)] S = S 1 =, contrary to our assumption that S is a plds of G + H Therefore, (iv) holds Similarly, (v) holds if S 1 = 1 and S

8 5338 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario Case S 1 = and S = Suppose that V (G) \ N[S 1 ] Then there exist x, y V (G) such that x, y / N[S 1 ] This implies that [N G+H (x) N G+H (y)] S = Thus, S does not satisfy the Definition of p-liar s (ii), contrary to our assumption that S is p-liar s dominating set of G + H Hence, V (G) \ N[S 1 ] 1 Similarly, V (H) \ N[S ] 1 Case 3 S 1 = and S 3 or S 1 3 and S = Suppose that S 1 = and S 3 and suppose further that V (H) \ N[S ] Then, [N G+H (x) N G+H (y)] S = S 1 = for some x, y V (H)\N[S ], contrary to the fact that S is a p-liar s dominating set of G+H Thus, V (H) \ N[S ] 1, showing that (vi) holds Similarly, (vii) holds if S 1 3 and S = The converse is clear As a consequence of Theorem 34, we have the next results Remark 35 Let G and H be non-trivial connected graphs Then 3 γ plr (G + H) 6 Corollary 36 Let G and H be non-trivial connected graphs Then γ plr (G + H) = 3 if and only if one of the following holds: (i) γ plr (G) = 3; (ii) γ plr (H) = 3; (iii) γ a (H) = 1 and γ (G), where γ (G) = min { S : S is a dominating set of G with epn(x; S ) 1 for all x S }; or (iv) γ a (G) = 1 and γ (H), where γ (H) = min { S 1 : S 1 is a dominating set of H with epn(x; S 1 ) 1 for all x S 1 } Corollary 37 Let G and H be non-trivial connected graphs such that γ plr (G + H) 3 Then γ plr (G + H) = 4 if and only if one of the following holds: (i) γ plr (G) = 4; (ii) γ plr (H) = 4; (iii) γ a (G) and γ a (H) ; (iv) γ (H) = 3, where γ (H) = min { S : S is a dominating set of H with epn(x; S ) 1 for all x S }; or

9 p-liar s domination, join and corona of graphs 5339 (v) γ (G) = 3, where γ (G) = min { S : S is a dominating set of G with epn(x; S ) 1 for all x S } Corollary 38 Let G and H be non-trivial connected graphs such that γ plr (G + H) > 4 Then γ plr (G + H) = 5 if and only if one of the following holds: (i) γ plr (G) = 5; (ii) γ plr (H) = 5; (iii) γ a (H) = 3; (iv) γ a (G) = 3; (v) γ (G) = 4, where γ (G) = min { S : S is a dominating set of G with epn(x; S ) 1 for all x S }; or (vi) γ (H) = 4, where γ (H) = min { S : S is a dominating set of H with epn(x; S ) 1 for all x S } 4 p-liar s Domination in the Corona of Graphs For every v V (G), denote by H v the copy of H whose vertices are attached one by one to the vertex v Denote by v +H v the subgraph of the corona G H corresponding to the join {v} + H v Theorem 41 Let G be non-trivial connected graph and H be any graph Then C V (G H) is a p-liar s dominating set of G H if and only if C = A ( v A S v ) ( u/ A D u ), where A V (G), S v is a dominating set of H v with epn(x; S v ) 1 for each x S v and for each v A, and D u is a p-liar s dominating set of H v for each u / A, where D u = V (H v ) whenever V (H) = or 3 Proof : Suppose C is a plds of G Let A = C V (G) Pick any v A and let S v = C V (H v ) By Theorem 31, S v is a dominating set of H v and epn(x; S v ) 1 for all x S v Next, let u / A and set D u = C V (H v ) By Theorem 31, D u is a p-liar s dominating set of H u, where D u = V (H v ) whenever V (H) = or 3 For the converse, suppose that C has the given form and the corresponding properties Let z V (G H) \ C and let w V (G) such that z V (w + H w ) Suppose z = w Then z / A By assumption, D z is a p-liar s dominating set of H z, where D z = V (H z ) whenever V (H) = or 3 Hence, N G H (z) C D z Suppose z w Then z V (H w ) \ C If w / A, then

10 5340 Carlito B Balandra, Sergio R Canoy, Jr, and Imelda S Aniversario D w = C V (H w ) is a p-liar s dominating set of H w Since z / D w, it follows that N G H (z) C = N H w(z) D w If w A, then S w = C V (H w ) is a dominating set of H w by assumption Thus, N G H (z) C 1 + N H w(z) S w Hence, C is a -dominating set of G H Finally, let a, b V (G H) \ C (a b) and let u, v V (G) such that a V (u + H u ) and b V (v + H v ) Consider the following cases: Case 1 u = v Suppose u A Then, by assumption, S u is a dominating set of H u with epn(x; S u ) 1 for all x S u Thus, since a, b V (H u ) \ S u, [N G H (a) N G H (b)] C = [N H u(a) N H u(b)] S u Suppose now that u / A Then D u = V (H u ) C is a p-liar s dominating set of H u, where D u = V (H u ) whenever V (H) = or 3 Suppose that one of a and b, say a, is u Then b V (H u ) \ D u Hence, by assumption, V (H) 4 This implies that D u 3 and [N G H (a) N G H (b)] D u 3 So suppose that a, b V (H u ) \ D u Since D u is a plds of H u, it follows that [N G H (a) N G H (b)] C = [N H u(a) N H u(b)] D u 3 Case u v Consider the following sub-cases: Sub-case 1 u, v A Then S u and S v are dominating sets of H u and H v, respectively, and a V (H u ) \ S u and b V (H v ) \ S v Thus, [N G H (a) N G H (b)] C = + N H u(a) S u + N H v(b)] S v 4 Sub-case u, v / A Then D u and D v are plds of H u and H v, respectively Thus, [N G H (a) N G H (b)] C 4 Sub-case 3 u A and v / A or u / A and v A Then, by assumptions, [N G H (a) N G H (b)] C 4 Accordingly, C is a plds of G H The next result is a direct consequence of Theorem 41 Corollary 4 Let G be non-trivial connected graph of order n 4 and H be any graph of order m Then γ plr (G H) =min{ A + A γ (H) + (n A )γ plr (H) : A V (G)}, where γ (H) = min { S v : S v is a dominating set of H v with epn(x; S v ) 1 for each x S v and for each v A} In particular, γ plr (G H) nγ plr (H)

11 p-liar s domination, join and corona of graphs 5341 References [1] TW Haynes, ST Hedetmiemi and PJ Slater, Fundamentals of Domination in Graphs, Vol 08 of Monographs and textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1998 [] M Nikodem, False Alarm in Fault-Tolerant Dominating Sets in Graphs, Applied Mathematics, AGH University of Science and Technology, al Mickiewicsa 30, Krakow, Poland [3] PJ Slater and ML Roden, Liar s Domination in Graphs, Discrete Mathematics, 309 (008), [4] C Sterling, Liar s Domination in Grid Graphs, Master of Science in Mathematical Sciences, East Tennesse State University, 01 Received: August 3, 015; Published: August 1, 015