A set S of vertices in a graph G is a double total dominating set, abbreviated DTDS, of G if every vertex of G is adjacent to least two vertices in S. The minimum cardinality of a DTDS of G is the double total domination number of G. In this paper, we study the DTDS of the generalized Petersen graphs. Mathematics Subject Classification: 05C35

A connected graph G is said to be (a, d)-antimagic, for some positive integers a and d, if its edges admit a labeling by the integers 1,2, ... , IE( G) I such that the induced vertex labels consist of an arithmetic progression with the first term a and the common difference d. In this paper we prove that the generalized Petersen graph P(n,2) is (3n2+6, 3)-antimagic for n == 0 (mod 4), n ~ B.

New results on singleton rainbow domination numbers of generalized Petersen graphs P(ck,k) are given. Exact values established for some infinite families, and lower upper bounds with small gaps given in all other cases.

Determining the size of minimum vertex cover of a graph G, denoted by β(G), is an NP-complete problem. Also, for only few families of graphs, β(G) is known. We study the size of minimum vertex cover in generalized Petersen graphs. For each 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...

A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 3-regular 4-ordered graphs other than K4 and K3,3. In 2008, Meszaros solve the question by proving the Petersen graph and the Heawood graph are 3-regular 4-ordered gr...

Let G = (V, E) be a graph. A subset S ⊆ V is a dominating set of G, if every vertex u ∈ V − S is dominated by some vertex v ∈ S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2...

We show that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or force the robber towards an end of the infinite graph. We prove that finite isometric subtrees are 1-guardable and apply this to determine the exact cop number of some famili...