Restrained reinforcement number in graphs

Authors

Kazhal Haghparast, Azarbaijan Shahid Madani UniversityFollow
Jafar Amjadi, Azarbaijan Shahid Madani UniversityFollow
Mustapha Chellali, University of BlidaFollow
Seyed Mahmoud Sheikholeslami, Azarbaijan Shahid Madani UniversityFollow

Publication Date

July 2022

Abstract

A set S of vertices is a restrained dominating set of a graph G=(V,E) if every vertex in V\ S has a neighbor in S and a neighbor in V\S. The minimum cardinality of a restrained dominating set is the restrained domination number γ_r(G). In this paper we initiate the study of the restrained reinforcement number r_r(G) of a graph G defined as the cardinality of a smallest set of edges F ⊆ E( ‾G) for which γ_r(G + F) < γ_r(G), where ‾G denotes the complement graph of G. We first show that the decision problem associated with r_r(G) is NP-hard for arbitrary graphs G. Then we establish various properties as well as some sharp bounds on the restrained reinforcement number.

Creative Commons License

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

Recommended Citation

Haghparast, Kazhal; Amjadi, Jafar; Chellali, Mustapha; and Sheikholeslami, Seyed Mahmoud (2022) "Restrained reinforcement number in graphs," Theory and Applications of Graphs: Vol. 9: Iss. 2, Article 9.
DOI: 10.20429/tag.2022.090209
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss2/9