gamma-Realizability and Other Musings on Inverse Domination

Authors

John Asplund, Dalton State CollegeFollow
Joe ChaffeeFollow
James M. Hammer III, Cedar Crest CollegeFollow
Matt Noble, Middle Georgia State UniversityFollow

Abstract

We introduce and study γ-realizable sequences. For a finite, simple graph G containing no isolated vertices, I ⊆ V(G) is said to be an inverse dominating set if I dominates all of G and I is contained by the complement of some minimum dominating set D. Define a sequence of positive integers (x₁,…, x_n) to be γ-realizable if there exists a graph G having exactly n distinct minimum dominating sets D₁,… D_n where for each i ∈ {1,… n}, the minimum size of an inverse dominating set in V(G) \D_i is equal to x_i. In this work, we show which sequences having minimum entry 2 or less are γ-realizable. We then detail a few observations and results arising during our investigations that may prove useful in future research.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Asplund, John; Chaffee, Joe; Hammer, James M. III; and Noble, Matt (2018) "gamma-Realizability and Other Musings on Inverse Domination," Theory and Applications of Graphs: Vol. 5: Iss. 1, Article 5.
DOI: 10.20429/tag.2018.050105
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol5/iss1/5