A set S ⊂ V is a dominating set of a graph G = (V, E) if every vertex υ ∈ V 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)?
@article{bwmeta1.element.doi-10_1515_amsil-2015-0003,
author = {Anna Bie\'n},
title = {Gamma Graphs Of Some Special Classes Of Trees},
journal = {Annales Mathematicae Silesianae},
volume = {29},
year = {2015},
pages = {25-34},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_amsil-2015-0003}
}
Anna Bień. Gamma Graphs Of Some Special Classes Of Trees. Annales Mathematicae Silesianae, Tome 29 (2015) pp. 25-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_amsil-2015-0003/
[1] Diestel R., Graph theory, Springer-Verlag, Heidelberg, 2005.
[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. | Zbl 1229.05219
[3] Haas R., Seyffarth K., The k-dominating graph, Graphs Combin. 30 (2014), 609–617.[Crossref] | Zbl 1294.05122
[4] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals on domination in graphs, CRC Press, New York, 1998. | Zbl 0890.05002
[5] Lakshmanan S.A., Vijayakumar A., The gamma graph of a graph, AKCE J. Graphs Combin. 7 (2010), 53–59. | Zbl 1223.05212