Abstract
Given a graph G, the k-dominating graph of G, D k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D k (G). In this paper we give conditions that ensure D k (G) is connected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berge, C.: Some classes of perfect graphs. In: Six Papers in Graph Theory. Indian Statistical Institute, McMillan, pp. 1–21 (1963)
Bonsma P., Cereceda L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci. 410(50), 5215–5226 (2009)
Bondy, J.A., Murty, U.S.R.: Graph Theory. GTM 244, Springer, Berlin (2008)
Cereceda L., Heuvel J., Johnson M.: Connectedness of the graph of vertex-colourings. Discrete Math. 308, 913–919 (2008)
Cereceda L., Heuvel J., Johnson M.: Finding paths between 3-colorings. J. Graph Theory 67, 69–82 (2011)
Choo K., MacGillivray G.: Gray code numbers for graphs. Ars Math. Contemp. 4, 125–139 (2011)
Fricke G., Hedetniemi S.M, Hedetniemi S.T., Hutson K.R.: γ-graphs of graphs. Discuss. Math. Graph Theory 31(3), 517–531 (2011)
Haas R.: The canonical coloring graph of trees and cycles. Ars Math. Contemp. 5, 149–157 (2012)
Haynes, T., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York (1998)
Ito, T., Kaminski, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. In: Algorithms and data structures. Lecture Notes in Computer Science, vol. 5664, pp. 375–386. Springer, Berlin, (2009)
Ito T., Demaine E.D., Harvey N.J.A., Papadimitriou C.H., Sideri M., Uehara R., Uno Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412, 1054–1065 (2011)
Jacobson M.S., Peters K.: Chordal graphs and upper irredundance, upper domination, and independence. Discrete Math. 86, 59–69 (1990)
Lovász L.: Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)
Subramanian K., Sridharan N.: γ-graph of a graph. Bull. Kerala Math. Assoc. 5(1), 17–34 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Haas, R., Seyffarth, K. The k-Dominating Graph. Graphs and Combinatorics 30, 609–617 (2014). https://doi.org/10.1007/s00373-013-1302-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-013-1302-3