ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 25(1-2) (2005) 121-128
DOI: 10.7151/dmgt.1266


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
Department of Computer Science
University of Veszprém
H-8200 Veszprém, Egyetem u. 10, Hungary


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.


[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] G. Bacsó and Zs. Tuza, Dominating cliques in P5-free graphs, Periodica Math. Hungar. 21 (1990) 303-308, doi: 10.1007/BF02352694.
[3] G. Bacsó and Zs. Tuza, Domination properties and induced subgraphs, Discrete Math. 1 (1993) 37-40.
[4] G. Bacsó and Zs. Tuza, Dominating subgraphs of small diameter, J. Combin. Inf. Syst. Sci. 22 (1997) 51-62.
[5] G. Bacsó and Zs. Tuza, Structural domination in graphs, Ars Combinatoria 63 (2002) 235-256.
[6] M.B. Cozzens and L.L. Kelleher, Dominating cliques in graphs, pp. 101-116 in [10].
[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.
[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, N.Y., 1998).
[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.
[10] - Topics on Domination (R. Laskar and S. Hedetniemi, eds.), Annals of Discrete Math. 86 (1990).

Received 3 November 2003
Revised 17 November 2004