Akademik Veri Yönetim Sistemi
RAIRO-OPERATIONS RESEARCH, cilt.59, sa.2, ss.1247-1256, 2025 (SCI-Expanded, Scopus)
Let G = (V, E) be a graph of order n. For S subset of V (G), the set Np(S) is defined as the perfect neighborhood of S such that all vertices in V (G)\S have exactly one neighbor in S. The perfect differential of S is defined to be partial derivative p(S) = |Np(S)| - |S| and the perfect differential of a graph is defined as partial derivative p(G) = max{partial derivative p(S) : S subset of V (G)}. A perfect Roman dominating function is defined as a Roman dominating function f satisfying the condition that every vertex u for which f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The perfect Roman domination number, denoted by gamma pR(G), is the minimum weight among all perfect Roman dominating functions on G, that is gamma pR(G) = min{w(f) : f is a perfect Roman dominating function on G}. Let G be the complement of a Graph G. The complementary prism GG of G is the graph formed from the disjoint union of G and G by adding the edges of a perfect matching between the corresponding vertices of G and G. This paper is devoted to the computation of perfect differentials of complementary prisms G G and perfect Roman domination numbers of complementary prisms G G by the use of the Gallai-type result proven before. Particular attention is given to the complementary prims of special types of graphs. Furthermore, a sharp lower bound on the perfect differential of the complementary prism G G of a graph G in terms of the order of G is presented and the graphs attaining this lower bound are characterized. Finally, the graphs are characterized for which partial derivative p(GG) and gamma pR(GG) are small.