Abstarct: Given a positive integer k≥2 , two vertices in a graph are said to ‎k-step dominate each other if they are at distance k apart. A set S of vertices in a graph G is a k-step dominating set of G if every vertex is k-step dominated by some vertex of S . The k-step domination number of ‎G is the minimum cardinality of a k-step dominating set of G. A subset S of vertices of G is a k-hop dominating set if every vertex outside S is ‎ k‎-step dominated by some vertex of ‎S. ‎In this thesis‎, we show that for any integer k≥2 , the decision problems for the ‎k-step dominating set and ‎k-hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs. ‎Also ‎we obtain‎ ‎new upper bounds on the size of exact 1-step domination graphs‎, and present an upper bound on the total domination number of ‎‎an exact 1-step domination tree and characterize trees achieving ‎‎equality for this bound‎.‎ Then we study on strongest dominating set in fuzzy graphs and obtain these strengths for fuzzy complete graphs and fuzzy complete bipartite graphs. Also‎ ‎we define exact k-Step dominating sets in fuzzy graphs and determine strongest exact k-Step dominating set for fuzzy graphs ‎and‎ obtain new upper bound on the size of exact 1-step domination fuzzy graphs‎.