A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 adjacent to at least one assigned 3 or two vertices 2, and with f(u)=1 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. We obtain closed expressions generalized Petersen graphs P(5k,k). It proven ?dR(P(5k,k))=8k k?2,3mod5 8k??dR(P(5...

A Roman domination function on a graph G = (V,E) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman domination function f is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by...

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for infinite families, exact values are established; in all other the lower and upper bounds with small gaps given. also define singleton rainbow domination, where sets assigned have a cardinality of, at most, one, provide analogous this special case domination.

We obtain new results on 2-rainbow domination number of generalized Petersen graphs P(5k,k). In some cases (for infinite families), exact values are established, and in all other lower upper bounds given. particular, it is shown that, for k>3, γr2(P(5k,k))=4k k≡2,8mod10, γr2(P(5k,k))=4k+1 k≡5,9mod10, 4k+1≤γr2(P(5k,k))≤4k+2 k≡1,6,7mod10, 4k+1≤γr2(P(5k,k))≤4k+3 k≡0,3,4mod10.

Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: K2ƒC2n, for some n ≥ 2, generalized Petersen graph G(10, 3), permutahedron, truncated cuboctahedron, or truncated icosidodecahedron. This complete classification of cubic, vertex-transitive partial cubes, a f...

For natural numbers n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and subscripts are reduced modulo n. Here first, we characterize minimum vertex covers in generalized Petersen graphs. Second, we present a lower bound and some upper bounds for β(P (n, k)), the size of minimum vertex cover...