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Total Domination Number of Generalized Petersen Graphs

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  • Intelligent Information Management, 2009, 15-18 doi:10.4236/iim.2009.11003 Published Online July 2009 (http://www.scirp.org/journal/iim)

    Copyright 2009 SciRes IIM

    Total Domination Number of Generalized Petersen Graphs

    Jianxiang CAO1, Weiguo LIN2, Minyong SHI3 1 School of Computer Science, Communication University of China, Beijing 100024, China 2 School of Computer Science, Communication University of China, Beijing 100024, China

    3 School of Animation, Communication University of China, Beijing 100024, China

    Abstract: Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied . The total domination number of generalized Petersen graphs P(m,2) is obtained in this paper.

    Keywords: generalized Petersen graphs, total domination set, total domination number, regular graph, domi- nation set, domination number

    Petersen

    1 2 3 1 1000242 100024

    3 100024

    Petersen Petersen

    Petersen

    1.

    NP-[1] De Bruijn Kauta

    Petersen Petersen Petersen

  • TOTAL DOMINATION NUMBER OF GENERALIZED PETERSEN GRAPHS 16

    [2] Petersen Hamiltonian [3] Petersen

    crossing number[4]Petersen P(m,2)[5][6] Petersen Petersen

    2. (graph)(V, E) V(vertex-set)(vertex) E(edge-set)(edge) V(G) E(G) G e(incident) e=(u, v) u, v V, u, v

    e(end-vertex) u, v(adjacent) .(u, v)=(v, u) (undirected graph) u vW V(G),W G, v V(G), N(v)={u V(G)| uv E(G)} N[v]={v} N(v); A V(G), N[A]={v|v A

    u A uv E(G)}; u u k k

    1 Petersen P(m, a),P(m, a) U W, U={u1,u2,,um}, W={w1,w2,,wm}; P(m, a) (ui, wi)(1im),(ui,ui+1)(i1,ui=uj, ij(mod m))(wi, wi+a)(i1,wi=wj, ij(mod m)) U

    W Petersen Coxter1951 Petersen P(5, 2)

    2 (u,v) G u v T V(G) G T G

    t (G)

    G x[7]

    x

    3. 1 G n k t (G)

    kn

    k T G u,v T uv N(u)N(v)= , T k

    T

    kn 1 Petersen P(m,2)

    t (P(m,2)) =

    k2

    23)1(21312

    3k2

    kmkkmk

    km

    P(m,2) Petersen

    1 m=3k t (P(3k,2))=2k, k=1,2, T={u1,w1,u4,w4,,u3k-2,w3k-2},|T|=2k, T P(3k,2)

    1.1 k P(3k,2) 3k/2 3k , C=u1u2u3u3ku1 C1C2

    C1=w1w3w5w3k-1w1;

    C2=w2w4w6w3kw2.

    (ui, wi)(1im) P(3k,2) T T u3i-2(1ik) u3k,u2, u3i-3 , u3i-1 2ik,

    T

    T C1 w1,w7,,w3k-5C1 w3k-1,w3,w5, ,w3k-4 T C2 w4,w10,,w3k-2 C2 w2,w6,w8, w10,,w3k T

    t (P(3k,2))|T|=2k P(3k,2) 3 1

    t (P(3k,2))(23k)/3=2k t (P(3k,2))=2k

    Copyright 2009 SciRes IIM

  • JIANXIANG CAO, WEIGUO LIN, MINYONG SHI 17

    1.2 k P(3k,2) 1.1 C 3k C1 =w1w3w5w3kw2w4w6w3k-1w1. 1.1T u3i-2(1ik) u3k,u2, u3i-3, u3i-1 2ik, T

    T C1 w1,w4,,w3k-2C1 w3k,w3, w5, ,w3k . T T

    t (P(3k,2))|T|=2k P(3k,2) 3 1 t

    (P(3k,2))(23k)/3=2k t (P(3k,2))=2k 1.1 1.2 m=3kk1 t (P(3k,2))=2k

    2 m=3k+1 , t (P(3k+1,2))=2(k+1), k=1,2,

    1.1 k=1 P(4,2) W={w1,w2,w3,w4}

    (w1,w3)(w2,w4) T P(4,2) P(4,2) 3 1 t (P(4,2))

    342 =3 t (P(4,2))=3, |T|=3T

    T 3 P(4,2) 3 4

    (1) 3W 3

    (2) T 3 2 T={u1,u2,w1},

    w4 (3) T 1 2W T={u1,w1,w3}, w2,w4

    (4) T 3 W W={w1,w2,w3,w4}

    3 4 P(4,2) 3 w1,w2,w3 w4

    P(4,2) t (P(4,2))=4=2(k+1) k2 k

    k 1.2 k

    P(3k+1,2) C C1

    C=u1u2u3u3ku1 ; C1 =w1w3w5w3k+1w2w4w6w3kw1 ; P(3k+1,2) 3 1 t

    (P(3k+1,2))

    3

    )13(2 k=2k+1

    T1={u1,w1,u4,w4,,u3k-2,w3k-2,u3k-1 ,u3k+1} T1 ui wi, u1 u3k+1

    u2,u3k+1 u3k T1 ui ui-1 ui+1 i=4,7,,3k-2w1w3k,w3, T1 wi wi-2 wi+2 i=4,7,,3k-2 u3k-1 u3k+1 w3k-1 w3k+1 T1

    P(3k+1,2)|T1|=2(k+1) t ( P ( 3 k + 1 , 2 ) ) 2 ( k + 1 ) , t (P(3k+1,2))

    3)13(2 k

    =2k+1 2k+1 t (P(3k+1,2))2(k+1)

    t (P(3k+1,2))2k+1 t (P(3k+1,2))=2(k+1)

    t (P(3k+1,2))=2k+1 T|T|=2k+1

    T T PetersenT 3

    3 3 2 3 3w3, w5, w7, 3 u3, u5, u7 w1, w9u3, u4, u5, u6, u7 u4, u6 w4, w6

    u5 w3, w5, w7, w4, w6 14 2(3k+1)-14 2(3k+1)-14/3=2k-4 T 2w11,,w3k+1(3k-8)/2

    Copyright 2009 SciRes IIM

  • TOTAL DOMINATION NUMBER OF GENERALIZED PETERSEN GRAPHS

    Copyright 2009 SciRes IIM

    18

    T 1 T 2k-4 w11,,w3k+1T

    (3k-8)/(23) T 2 3 2 3 u3,w3,w5 3 u2, u3, u4, w1, w3, w5, w7 3k+1-3+3k+1-4=6k-5, 3 36k-5 2k-1

    2k-1+3=2k+2 |T|=2k+1 2k+1 P(3k+1,2) t (P(3k+1,2))=2(k+1)

    1.3 k 1.2

    t (P(3k+1,2))=2(k+1) 2

    3 m=3k+2k1 1 t ( P ( 3 k + 2 , 2 ) )

    3

    )23(2 k =2k+2

    t (P(3k+2,2))=2k+2 T V(P(3k+2,2)),|T|=2k+2

    1.1 k P(3k+2,2) 3k+2 C=u1u2u3u3k+2u1P(3k+2,2) 3k/2 C1C2

    C1=w1w3w5w3k+1w1; C2=w2w4w6w3k+2w2.

    T={u2, w2, u5, w5,, u3k-1, w3k-1, u3k, u1}, T ui wi i=2,5,,3k-1. ui ui-1 ui+1u3k u3k+1u1u3k+2 T T w3i+2 C1 w3i w3i+4i=1,3,,k-1 u1 w1 C1

    T C2T w2 C2 w3k+2 w4 ,Tw3i+2

    C2 w3i w3i+4 i=2,4,,k-2 u3k w3k C2 T T P(3k+2,2)|T|=2k+2 T t (P(3k+2,2))= 2(k+1)

    1.2 k P(3k+2,2) C=u1u2u3u3k+ 2u1 3k C1 =w1w3w5w3k+2w2w4w6w3k+1w1

    T={u2, w2, u5, w5, , u3k-1, w3k-1, u3k, u1}, 1.1 T P(3k+2,2) t (P(3k+2,2))=2(k+1)

    m=3k+2 t (P(m,2))=2(k+1)

    REFERENCES [1] Huang Jia, On Domination-Stability of Graphs. The doctoral

    dissertation of the Univerisity of Science and Technology og China.(in Chinese)

    (. 2007). [2] Bnaani K. Hamiltonian cycles in generalized Petersen graphs [J].

    J Combinatorial Theory, 1978, 181-183. [3] Exoo G, Harary F, Kabell J. The crossing numbers of some

    generalized Petersen graphs [J]. Math Scand, 1981, 184-188. [4] Hou Xin-min, Wang Tian-ming. Wide diameters of generalized

    Petersen graphs [J]. Journal of Mathematical Research and Exposition, 2004, 24(2): 249-253.

    [5] QI Deng-ji, Domination Number of Generalized Petersen Graphs (m is odd) , Journal of Qingdao University of Science and Technology , Vol . 26 No. 1 Apr . 2005, 92-94(in Chinese) (. Petersen m P(m,2)[J]. , 2005, 26(1): 92-94).

    [6] QI Deng-ji, Domination Number of Generalized Petersen Graphs (m is even) , Journal of Qingdao University of Science and Technology , Vol . 26 No. 2 Apr . 2005, 181-183(in Chinese)(. Petersen m P(m,2)[J]. , 2005, 26(2): 181-183).

    [7] Bondy J A, Murty U S R. Graph theory with applications [M]. New York: North-Holland, 1976.