# Graph domination in distance two

• Volume: 25, Issue: 1-2, page 121-128
• ISSN: 2083-5892

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## Abstract

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Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that $DomDo{m}_{u}=Dom{₂}_{u}$ holds for ${}_{u}$ = all connected graphs without induced ${P}_{u}$ (u ≥ 2). (In particular, ₂ = K₁ and ₃ = all complete graphs.) Some negative examples are also given.

## How to cite

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Gábor Bacsó, Attila Tálos, and Zsolt Tuza. "Graph domination in distance two." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 121-128. <http://eudml.org/doc/270765>.

@article{GáborBacsó2005,
abstract = {Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that $Dom Dom _u = Dom₂ _u$ holds for $_u$ = all connected graphs without induced $P_u$ (u ≥ 2). (In particular, ₂ = K₁ and ₃ = all complete graphs.) Some negative examples are also given.},
author = {Gábor Bacsó, Attila Tálos, Zsolt Tuza},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph; dominating set; connected domination; distance domination; forbidden induced subgraph; -dominating subgraph},
language = {eng},
number = {1-2},
pages = {121-128},
title = {Graph domination in distance two},
url = {http://eudml.org/doc/270765},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Gábor Bacsó
AU - Attila Tálos
AU - Zsolt Tuza
TI - Graph domination in distance two
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 1-2
SP - 121
EP - 128
AB - Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that $Dom Dom _u = Dom₂ _u$ holds for $_u$ = all connected graphs without induced $P_u$ (u ≥ 2). (In particular, ₂ = K₁ and ₃ = all complete graphs.) Some negative examples are also given.
LA - eng
KW - graph; dominating set; connected domination; distance domination; forbidden induced subgraph; -dominating subgraph
UR - http://eudml.org/doc/270765
ER -

## References

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1. [1] G. Bacsó and Zs. Tuza, A characterization of graphs without long induced paths, J. Graph Theory 14 (1990) 455-464, doi: 10.1002/jgt.3190140409.
2. [2] G. Bacsó and Zs. Tuza, Dominating cliques in P₅-free graphs, Periodica Math. Hungar. 21 (1990) 303-308, doi: 10.1007/BF02352694.
3. [3] G. Bacsó and Zs. Tuza, Domination properties and induced subgraphs, Discrete Math. 1 (1993) 37-40.
4. [4] G. Bacsó and Zs. Tuza, Dominating subgraphs of small diameter, J. Combin. Inf. Syst. Sci. 22 (1997) 51-62.
5. [5] G. Bacsó and Zs. Tuza, Structural domination in graphs, Ars Combinatoria 63 (2002) 235-256.
6. [6] M.B. Cozzens and L.L. Kelleher, Dominating cliques in graphs, pp. 101-116 in [10]. Zbl0729.05043
7. [7] P. Erdős, M. Saks and V.T. Sós Maximum induced trees in graphs, J. Combin. Theory (B) 41 (1986) 61-79, doi: 10.1016/0095-8956(86)90028-6. Zbl0603.05023
8. [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, N.Y., 1998). Zbl0890.05002
9. [9] E.S. Wolk, The comparability graph of a tree, Proc. Amer. Nath. Soc. 3 (1962) 789-795, doi: 10.1090/S0002-9939-1962-0172273-0. Zbl0109.16402
10. [10] - Topics on Domination (R. Laskar and S. Hedetniemi, eds.), Annals of Discrete Math. 86 (1990).

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