Graph domination in distance two

• Autorzy: Gábor Bacsó , Attila Tálos , Zsolt Tuza
• Języki publikacji: EN

Abstrakty

EN

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, Domₖ 𝓓 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 $Dom Dom 𝓓_u = Dom₂ 𝓓_u$ holds for $𝓓_u$ = {all connected graphs without induced $P_u$} (u ≥ 2). (In particular, 𝓓₂ = {K₁} and 𝓓₃ = {all complete graphs}.) Some negative examples are also given.

Słowa kluczowe

EN

graph; dominating set; connected domination; distance domination; forbidden induced subgraph

Kategorie tematyczne

• 05C12: Distance in graphs
• 05C75: Structural characterization of families of graphs
• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2005
• Tom: 25
• Numer: 1-2
• Strony: 121-128

Daty

wydano

2005

otrzymano

2003-11-03

poprawiono

2004-11-17

Twórcy

autor

Gábor Bacsó

• Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary

autor

Attila Tálos

• Eötvös Lóránd University, H-1088 Budapest, Múzeum krt. 6-8, Hungary

autor

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

Bibliografia

• [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).

Identyfikatory

DOI

10.7151/dmgt.1266