Discussiones Mathematicae Graph Theory 25(1-2) (2005)
121-128
DOI: https://doi.org/10.7151/dmgt.1266
GRAPH DOMINATION IN DISTANCE TWO
Gábor Bacsó
Computer and Automation Institute |
Attila Tálos
Eötvös Lóránd University |
Zsolt Tuza
Computer and Automation Institute |
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.
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Received 3 November 2003
Revised 17 November 2004
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