Note: 2-connected graphs with small 2-connected dominating sets

Authors:
Yair Caro;Raphael Yuster
Affiliations:
Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel;Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel
Venue:
Discrete Mathematics
Year:
2003

Citing 2
Cited 2

The probabilistic method

SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Connected Domination and Spanning Trees with Many Leaves

SIAM Journal on Discrete Mathematics

Probabilistic analysis of upper bounds for 2-connected distance k-dominating sets in graphs

Theoretical Computer Science
Note: On the (h,k)-domination numbers of iterated line digraphs

Discrete Applied Mathematics

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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 that γ2(G) ≤ n ln δ(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).