Signed Roman Domination Number of Generalized Petersen Graph

Shailaja S. Shirkol, Manjula C. Gudgeri

Abstract: A signed Roman Dominating Function (SRDF) on a graph G is a function f: V (G) {-1, 1, 2} such that _ (uN|V|) f (u) 1 for every v V (G) and every vertex u V (G) for which f (u) = -1 is adjacent to at least one vertex w for which f (w) = 2. The weight of SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. For natural number n and k, where n > 2k, a generalized Petersen graph P (n, k) is obtained by letting its vertex set to be {u1, u2, , un, v1, v2, , vn} and its edge set to be {ui ui+1, ui vi, vi vi+k}; where i = 1, 2,. . . , n and subscripts are reduced modulo n. In this paper we determine the signed Roman domination number of generalized Petersen graph P (n, k) for k = 1 & 3. We characterize generalized Petersen graph which have efficient signed Roman domination number.

Keywords: Generalized Petersen graph, Roman domination, Signed domination, signed Roman dominating function, signed Roman domination number & efficient signed Roman domination