Lower Bounds and Algorithms for Dominating Sets in

Web Graphs

Colin Cooper, Ralf Klasing, Michele Zito

To cite this version:

Colin Cooper, Ralf Klasing, Michele Zito. Lower Bounds and Algorithms for Dominating Setsin Web Graphs. [Research Report] RR-5529, INRIA. 2006, pp.26. <inria-00070478>

HAL Id: inria-00070478

https://hal.inria.fr/inria-00070478

Submitted on 19 May 2006

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Lower Bounds and Algorithms for Dominating Setsin Web Graphs

Colin Cooper — Ralf Klasing — Michele Zito

N° 5529

Mars 2005

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

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t (1 − Pt

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wDkxu ©x_*t._)q*rx]\slm_)moZ\_£vwzx_pkmdkosqZmZw%m

(

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2

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GC,mt

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(H H" !d!"Hi33P!"" H !

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Srx9kou^_

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ku tLmZ\_(£ ut_^wzKrxzl_*z1, ..., 2n

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k¡NmoZ\_a£_*Ém

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uu ka1 + 1

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xkmoZ_

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(v, u)£ _m(mZ\_

(k − 1)¥=moZ zuDZTm_*tlNrxu tTm(ZwX©D_-£ wx¦_^£

2(k− 1) + s+ 2wxt·u?mZ\_*z_Kuvkw

(v, u)_^lD_K£ _mumZwX©x_¨£vw¦N_*£

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FLu k(moZ\_tjs\](¦N_*z7rK£_*Ém7wxznmu wx£wxuzu t\Dk(r

2(k +1) + s+ a1 + a2

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x ]c324 ^3 gS

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£ wx¦_^£ k^§ t_*umoZ\_^z¨q*wDkn_

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(

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s+ a1 + a2

)

(s+ a1 + a2)!Φ(2(n− ((k+ 1) + s+ a1 + a2))).? C.A

moZ_*z_euvkct\r(v, u)

_^\x_emoZ\_^t

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(

2k + s− 4

s

)

Φ(2(k − 2)),?;RA

wxtu­moZ\_^zo_uvkcw(v, u)

_^lD_KmZ\_*t

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(

2k + s− 4

s

)

Φ(2(k − 2)).?I5A

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(v, u)u tGmZ\_*tL_*«jzo_pkokourDtka? C<A¤¡

?=NA.wz_)s\tqZwxt\x_prDtkow\\u t\wt (G) =

(φ(G))

§ moZ\_^zo_uvkcwt_^\x_(v, u)

moZ\_^t

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?=NAcZ_*z_

cuvkxu ©x_*tªwx¦r%©D_x§°V_kn_^_~mZw%m"u­

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§ tzo_^wzw%murDt·®rxzmZ\uvkcB_)t\rxmo_RQ?_m

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GZwxkcwmc£_pwxknm7_plx_pk

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(xi+1, xi)wxt"w%mc£ _^wxknm¨a_^lD_^k

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(S dom. S) ≤ (S ∪ [σ] dom. S \ [σ]).

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wDk[σ]

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Au = G ∈ : ∃(v, u)_plx_)u t

G ∩ (BI ∪ BII),

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(v, u).YZ_\zrTrx®rDz

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S\rx]utw%mo_pk

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A§

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x > vknrmoZw%m

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u tq*£s\_wxt_^lD_(®zrx]x§'=£vknr

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(φ(G(α1, α2))).

?;RA

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(G(b2, b1))

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§ moZ\_^zo_)uvkcw(v, u)

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j

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j

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m ≥ 2§

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(x(1), ..., x(m))rxkos\¦l¥´©x_^znmu q*_^krxl_^xz_*_

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1, ..., 2mn§¬_*mmoZ\_

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eiwt

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u(j + k)®rxzkow\\u t\§

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t ¦N_amoZ\_-kowxq*_(rkoq^w£ _¥=®zo_^_xzwZk¨rxrxs\mn¥Øl_*Dzo_^_ m §aYZ\_7kowDq_GS

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ej , j = 1, ...,mrx?©x_^znm_«

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v1, ..., vt−1wz_)utq*£sl_^¢§QYZjsku tmoZ\_kq*wx£_*¥;®z_*_e]rll_*£

(ej

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2m(t− 1)

2m(t− 1) + 2(j − 1) + 1j = 1, ...,m.

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ÿÕ¬ÿÉü

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399