The Study of Secure-dominating Set of Graph Products

Transcription

1 The Study of Secure-dominating Set of Graph Products Hung-Ming Chang Department of Mathematics National Kaohsiung Normal University Advisor: Hsin-Hao Lai August, 014

2 Outline 1 Introduction Preliminary and known results 3 Results on the strong product of two graphs 4 Results on lexicographic product 5 Conclusions 6 Bibliography

3 Secure Set An Attack on S A : S P(V (G) S) such that A(u) N G [u] S for any u S and A(u) A(v) = for any u v S. A Defense of S D : S P(S) such that D(u) N G [u] S for any u S and D(u) D(v) = for any u v S.

4 Secure Set An Attack on S A : S P(V (G) S) such that A(u) N G [u] S for any u S and A(u) A(v) = for any u v S. A Defense of S D : S P(S) such that D(u) N G [u] S for any u S and D(u) D(v) = for any u v S.

5 Secure Set An Attack on S A : S P(V (G) S) such that A(u) N G [u] S for any u S and A(u) A(v) = for any u v S. A Defense of S D : S P(S) such that D(u) N G [u] S for any u S and D(u) D(v) = for any u v S. R. Brigham, R. Dutton and S. Hedetniemi, 004 A subset S of V (G) is a secure set of G if for any attack A on S, there exists a defense D of S such that D(u) A(u) for any u S.

6 Secure Set A subset S of V (G) is a dominating set if N G [S] = V (G). Secure-dominating Set and Secure-dominating Number A subset S of V (G) is a secure-dominating set of G if S is a secure set of G that is also a dominating set of G. The secure-dominating number γ s (G) of G is the minimum cardinality of secure-dominating sets of G.

7 Secure Set A subset S of V (G) is a dominating set if N G [S] = V (G). Secure-dominating Set and Secure-dominating Number A subset S of V (G) is a secure-dominating set of G if S is a secure set of G that is also a dominating set of G. The secure-dominating number γ s (G) of G is the minimum cardinality of secure-dominating sets of G.

8 Preliminary Proposition.1 If S is a secure set of a graph G, then for each vertex v in S, N G [v] S N G (v) S. Theorem.4 For any graph G, γ s (G) G. Let S be a secure-dominating set. S D(u) A(u) V (G) S. u S u S

9 Known Results C.-L. Chang, T.-P. Chang and D. Kuo, 009 γ s (P m P n ) = mn, if m and n are at least two; γ s (P m C n ) = mn, if m and n 3; γ s (C m C n ) = mn + 1, if m (mod 4) and n 3 (mod 4); γ s (C m C n ) = mn, if m (mod 4) or n 3 (mod 4). γ s (K m1,m,...,m l ) = m 1+m + +m l, if l. K.-P. Huang and S.-T. Juan, 011 If l is an integer at least and m 1, m,..., m l are positive integers, then γ S (P m1 P m P ml ) = γ S (K m1 K m K ml ) = m 1 m m l.

10 Main Idea 1 Let V 1, V,..., V k be a partition of V (G). If S i is a secure-dominating set of G[V i ] for each 1 i k, and N[S i ] N[S j ] =, for any 1 i j k, then S 1 S S k is a secure-dominating set of G. If u is not in S, then u V i S for some i. Only the vertices in S i can be attacked by u.

11 Main Idea Let S be a dominating set and V (G) S = {v 1, v,..., v k }. If S can be partitioned into S 1, S,..., S k such that, for each i, N(v i ) S N[S i ] S. Then S is a secure-dominating set. If some vertex in S is attacked by v i, then we can use some vertex in S i to defense the attack.

12 Results on the strong product of two graphs Strong Product of Graphs Let X and Y be two graphs. The strong product X Y of X and Y is the graph such that V (X Y ) = V (X) V (Y ), (x 1, y 1 ) (x, y ) in X Y if and only if x 1 = x or x 1 x in X, and y 1 = y or y 1 y in Y.

13 P 9 P 5 S 9,5 = {(, j) : j 1, (mod 4)} {(i, j) : i 0, 3 (mod 4), 1 j 5}.

14 P 9 P 5 S 9,5 = {(, j) : j 1, (mod 4)} {(i, j) : i 0, 3 (mod 4), 1 j 5}.

15 P 9 P 5 S 9,5 = {(, j) : j 1, (mod 4)} {(i, j) : i 0, 3 (mod 4), 1 j 5}.

16 Results on the strong product of two graphs Lemma Let G and H are two graphs. If S G is a secure-dominating set of G, then S = {(s, h) : s S G, h V (H)} is a secure-dominating set of G H.

17 Results on the strong product of two graphs Lemma Let G and H are two graphs. If S G is a secure-dominating set of G, then S = {(s, h) : s S G, h V (H)} is a secure-dominating set of G H.

18 Results on the strong product of two graphs Lemma Let G and H are two graphs. If S G is a secure-dominating set of G, then S = {(s, h) : s S G, h V (H)} is a secure-dominating set of G H.

19 Results on the strong product of two graphs Lemma Let G and H are two graphs. If S G is a secure-dominating set of G, then S = {(s, h) : s S G, h V (H)} is a secure-dominating set of G H.

20 Results on the strong product of two graphs Theorem 6. Let G and H be two gaphs. We have γ s (G H) min{γ s (G) H, G γ s (H)}. Let S G be a secure-dominating set of G with size γ s (G) and S H be a secure-dominating set of H with size γ s (H). By Lemma, S = {(s, h) : s S G, h V (H)} and S = {(s, h ) : s V (G), h S H } are both secure-dominating sets of G H. Hence, γ s (G H) min{ S, S } = min{ S G H, G S H } = min{γ s (G) H, G γ s (H)}.

21 Results on the strong product of two graphs Corollary 6.3 Let G and H be two graphs. If γ s (G) = G, then γ s (G H) = G H. If γ s (G) = G, then γs (G H) min{γ s (G) H, G γ s (H)} = min{ G H, G γ s (H)} = G H = G H by Theorem.4. By Theorem 6., γ s (G H) G H. Hence, γ s (G H) = G H.

22 Results on H[G] Lexicographic Product of Graphs Let X and Y be two graphs. The lexicographic product X[Y ] of X and Y is the graph such that V (X[Y ]) = V (X) V (Y ), (x 1, y 1 ) (x, y ) in X[Y ] if and only if either x 1 x in X, or x 1 = x and y 1 y in Y.

23 Results on H[G] Lemma 7.1 Let G and H be two graphs. If S G is a secure-dominating set of G, then H[S G ] = {(h, g) : h V (H), g S G } is a secure-dominating set of H[G].

24 Results on H[G] Lemma 7.1 Let G and H be two graphs. If S G is a secure-dominating set of G, then H[S G ] = {(h, g) : h V (H), g S G } is a secure-dominating set of H[G].

25 Results on H[G] Lemma 7.1 Let G and H be two graphs. If S G is a secure-dominating set of G, then H[S G ] = {(h, g) : h V (H), g S G } is a secure-dominating set of H[G].

26 Results on H[G] Lemma 7.1 Let G and H be two graphs. If S G is a secure-dominating set of G, then H[S G ] = {(h, g) : h V (H), g S G } is a secure-dominating set of H[G].

27 Results Theorem 7. Let G and H be two graphs, we have γ s (H[G]) γ s (G) H. Let S G is a secure-dominating set with size γ s (G), then γ s (H[G]) H[S G ] γ s (G) H.

28 Conclusions γ s (P m P n ) = mn ; γ s (P m C n ) = mn, if m is even or n (mod 4); mn γs (P m C n ) mn + 1, if m is odd and n (mod 4); γ s (C m C n ) = mn, if m 0 (mod 4), or m and n are both odd mn except m n 3 (mod 4); γs (C m C n ) mn + 1, if m n 3 (mod 4) except m = n = 7, or m (mod 4) and n is odd; γs (C m C n ) mn +, if m and n are both 7 or m n (mod 4). mn

29 Conclusions γ s (G H) min{γ s (G) H, G γ s (H)}, if G and H are two graphs; γ s (P m G) = γ s (K m G) = m G, if m is even; m G γs (P m G) γ s (G) + (m 1) G, if m is odd; m G γs (K m G) γ S (G) + (m 1) G, if m is odd. γ s (H[G]) γ s (G) H for any two graphs G and H.

30 Bibliography 1. R. Aharoni, E. Milner and K. Prikry, Unfriendly partitions of a graph, J. Combin. Theory Ser. B 50 (1990), R. Brigham, R. Dutton and S. Hedetniemi, A sharp lower bound on the powerful alliance number of C m C n, Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 167 (004), R. Brigham, R. Dutton and S. Hedetniemi, Security in graphs, Discrete Appl. Math. 155 (007), R. Brigham, R. Dutton, T. Haynes and S. Hedetniemi, Powerful alliances in graphs, Discrete Math. 309 (009),

31 Bibliography 5. C.-L. Chang, T.-P. Chang and D. Kuo, Secure and secure-dominating set of graphs, manuscript (009). 6. Y.-H. Chang, Secure and secure-dominating sets of graphs, master s dissertation, National Dong Hwa University, Hualien, Taiwan (009). 7. K.-P. Huang, Secure and Secure-dominating set of Cartesian product graphs, master s dissertation, National Chai Nan University, Nantou, Taiwan (011). 8. P. Kristiansen, S. Hedetniemi and S. Hedetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (004), D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ 1996.