# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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# Discussiones Mathematicae Graph Theory

## GRAPH DOMINATION IN DISTANCE TWO

 Gábor Bacsó Computer and Automation Institute Hungarian Academy of Sciences H-1111 Budapest, Kende u. 13-17, Hungary Attila Tálos Eötvös Lóránd University H-1088 Budapest, Múzeum krt. 6-8, Hungary Zsolt Tuza Computer and Automation Institute Hungarian Academy of Sciences H-1111 Budapest, Kende u. 13-17, Hungary and Department of Computer Science University of Veszprém H-8200 Veszprém, Egyetem u. 10, Hungary

## 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 D of graphs, DomkD is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ D which is also connected. In our notation, DomD coincides with Dom1D. In this paper we prove that DomDomDu = Dom2Du holds for Du = {all connected graphs without induced Pu} (u ≥ 2). (In particular, D2 = {K1} and D3 = {all complete graphs}.) Some negative examples are also given.

Keywords: graph, dominating set, connected domination, distance domination, forbidden induced subgraph.

2000 Mathematics Subject Classification: 05C69, 05C75, 05C12.

## References

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Received 3 November 2003
Revised 17 November 2004