Roman type domination numbers of Fibonacci cubes

Interconnection networks have been developed for the communication needs of multiple computers. Interconnection networks can be represented by a graph G = (V(G),E(G)) with vertex set V(G) and edge set E(G). Here, the set V(G) represents the processors and the set E(G) represents the communication networks. One of the most basic interconnection models is the n-dimensional hypercube, Qn. While the vertices of this graph are labeled with all binary strings of length n, the edge set is formed by combining vertices with only one coordinate different. The n Fibonacci cube Γn is obtained by deleting all vertices containing a consecutive 1 from the vertex set of Qn. Let G be a graph and D be a subset of V, which is the vertex set of graph G. If each vertex in V is adjacent to an element of D or an element of D, the set D is called the domination set. The domination sets with the least number of elements are called the minimal domination set and the number of elements of this set is called the domination number of the G graph. Some domination type numbers of Fibonacci cubes are known in the literature. However, the Roman domination problem, which was developed as a war strategy and can be applied to the resource sharing problem, has not been studied in the literature for Fibonacci cubes. In this thesis, three Roman type domination numbers, namely Roman domination number, weak Roman domination number and double Roman domination number are discussed. By solving integer linear programming problems, these domination numbers are calculated as n ≤ 10 for Fibonacci cubes and the best lower and upper bounds are obtained for 11 ≤ n ≤ 13 dimensions.