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# Annales Mathematicae Silesianae

Editor-in-Chief: Sablik, Maciej

1 Issue per year

Mathematical Citation Quotient (MCQ) 2016: 0.10

Open Access
Online
ISSN
2391-4238
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Volume 29, Issue 1

# Gamma Graphs Of Some Special Classes Of Trees

Anna Bień
Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/amsil-2015-0003

## Abstract

A set SV is a dominating set of a graph G = (V, E) if every vertex υV which does not belong to S has a neighbour in S. The domination number γ(G) of the graph G is the minimum cardinality of a dominating set in G. A dominating set S is a γ-set in G if |S| = γ(G).

Some graphs have exponentially many γ-sets, hence it is worth to ask a question if a γ-set can be obtained by some transformations from another γ-set. The study of gamma graphs is an answer to this reconfiguration problem. We give a partial answer to the question which graphs are gamma graphs of trees. In the second section gamma graphs γ.T of trees with diameter not greater than five will be presented. It will be shown that hypercubes Qk are among γ.T graphs. In the third section γ.T graphs of certain trees with three pendant vertices will be analysed. Additionally, some observations on the diameter of gamma graphs will be presented, in response to an open question, published by Fricke et al., if diam(T (γ)) = O(n)?

(2010) Mathematics Subject Classification:: 05C69

Key words and phrases:: dominating set; gamma graph

## References

• [1] Diestel R., Graph theory, Springer-Verlag, Heidelberg, 2005.Google Scholar

• [2] Fricke G.H., Hedetniemi S.M., Hedetniemi S.T., Hutson K.R., γ-graphs of graphs, Discuss. Math. Graph Theory 31 (2011), 517–531.Google Scholar

• [3] Haas R., Seyffarth K., The k-dominating graph, Graphs Combin. 30 (2014), 609–617.

• [4] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals on domination in graphs, CRC Press, New York, 1998.Google Scholar

• [5] Lakshmanan S.A., Vijayakumar A., The gamma graph of a graph, AKCE J. Graphs Combin. 7 (2010), 53–59.Google Scholar

Revised: 2015-06-04

Published Online: 2015-09-30

Published in Print: 2015-09-01

Citation Information: Annales Mathematicae Silesianae, Volume 29, Issue 1, Pages 25–34, ISSN (Online) 0860-2107,

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