Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Strength of strongest dominating sets in fuzzy graphs2872971416310.22049/cco.2021.26988.1174ENMarziehFarhadi JalalvandShahrood University of TechnologyNaderJafari RadShahed UniversityMaryamGhoraniFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, iranJournal Article20201030A set <em>S</em> of vertices in a graph <em>G=(V,E)</em> is a dominating set of <em>G</em> if every vertex of <em>V-S</em> is adjacent to some vertex of <em>S</em>. For an integer <em>k≥1</em>, a set <em>S</em> of vertices is a <em>k</em>-step dominating set if any vertex of $G$ is at distance <em>k</em> from some vertex of <em>S</em>. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongest dominating set as well as strength of strongest $k$-step dominating set in fuzzy graphs. We determine various bounds for these parameters in fuzzy graphs. We also determine the strength of strongest dominating set in some families of fuzzy graphs including complete fuzzy graphs and complete bipartite fuzzy graphs. http://comb-opt.azaruniv.ac.ir/article_14163_39e9cbd30c7e18a6c6c3017a77610dd4.pdf