2-connected graphs with small 2-connected dominating sets

Yair Caro; Raphael Yuster

doi:10.1016/S0012-365X(03)00055-4

2-connected graphs with small 2-connected dominating sets

Yair Caro, Raphael Yuster

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a 2-connected graph. A subset D of V(G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ₂(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove thatγ₂(G)≤n lnδ(G)/δ(G) (1+o_δ(1)) where o_δ(1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3-8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202-211, Lovász, Discrete Math. 13 (1975) 383-390 and Payan, Cahièrs Centre Etudes Rech. Opér. 17 (1975) 307-317).

Original language	English
Pages (from-to)	265-271
Number of pages	7
Journal	Discrete Mathematics
Volume	269
Issue number	1-3
DOIs	https://doi.org/10.1016/S0012-365X(03)00055-4
State	Published - 28 Jul 2003

Keywords

Connectivity
Domination
Minimum degree

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

Access to Document

10.1016/S0012-365X(03)00055-4

Cite this

@article{1d664e7377e449e4a0fe045388ba5e9d,

title = "2-connected graphs with small 2-connected dominating sets",

abstract = "Let G be a 2-connected graph. A subset D of V(G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove thatγ2(G)≤n lnδ(G)/δ(G) (1+oδ(1)) where oδ(1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3-8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202-211, Lov{\'a}sz, Discrete Math. 13 (1975) 383-390 and Payan, Cahi{\`e}rs Centre Etudes Rech. Op{\'e}r. 17 (1975) 307-317).",

keywords = "Connectivity, Domination, Minimum degree",

author = "Yair Caro and Raphael Yuster",

year = "2003",

month = jul,

day = "28",

doi = "10.1016/S0012-365X(03)00055-4",

language = "English",

volume = "269",

pages = "265--271",

journal = "Discrete Mathematics",

issn = "0012-365X",

publisher = "Elsevier",

number = "1-3",

}

TY - JOUR

T1 - 2-connected graphs with small 2-connected dominating sets

AU - Caro, Yair

AU - Yuster, Raphael

PY - 2003/7/28

Y1 - 2003/7/28

N2 - Let G be a 2-connected graph. A subset D of V(G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove thatγ2(G)≤n lnδ(G)/δ(G) (1+oδ(1)) where oδ(1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3-8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202-211, Lovász, Discrete Math. 13 (1975) 383-390 and Payan, Cahièrs Centre Etudes Rech. Opér. 17 (1975) 307-317).

AB - Let G be a 2-connected graph. A subset D of V(G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove thatγ2(G)≤n lnδ(G)/δ(G) (1+oδ(1)) where oδ(1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3-8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202-211, Lovász, Discrete Math. 13 (1975) 383-390 and Payan, Cahièrs Centre Etudes Rech. Opér. 17 (1975) 307-317).

KW - Connectivity

KW - Domination

KW - Minimum degree

UR - http://www.scopus.com/inward/record.url?scp=0038817931&partnerID=8YFLogxK

U2 - 10.1016/S0012-365X(03)00055-4

DO - 10.1016/S0012-365X(03)00055-4

M3 - Article

AN - SCOPUS:0038817931

SN - 0012-365X

VL - 269

SP - 265

EP - 271

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1-3

ER -

2-connected graphs with small 2-connected dominating sets

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this