Roman Domination Number of the Cartesian Products of Paths and Cycles

Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from {0, 1, 2}. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give O(C) time algorithm for computing Roman domination number of special classes of polygraphs (rota– and fasciagraphs). By implementing the algorithm we give formulas for Roman domination number of the Cartesian products of paths and cycles Pn2Pk, Pn2Ck for k ≤ 8 and n ∈ N and for Cn2Pk and Cn2Ck for k ≤ 5, n ∈ N. We also give a list of Roman graphs among investigated families.