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Figure dominated more active in recent years, graph theory, a field of study, there are many practical applications in network design. Example, be placed on some of the nodes of a communication network transmitter, requiring each emitter node must have a direct communication line and an emitter node. How to select nodes such that the number of emitters placed a minimum, that is, a dominant problem. Calculation Figure dominated problem is NP-complete problem, so far only a minority class diagram disposable number is found and proved. Dominating set refers to a collection of S (?) V (G) any vertex v ∈ V (G) has a v ∈ S, or v and a point w adjacent and w ∈ S. That is, S is a dominating set if and only if N [S] = V (G). If the domination of any proper subset of the set S is not a dominating set of graph G, then dominating set S for a graph G is a minimal dominating set (minimal dominating set). If any graph G a minimal dominating set S *, meet | S ~~ * | ≥ | S |, called minimal dominating set S of a graph G is a minimum dominating set (minimum dominating set). The number of elements contained in the minimum dominating set S is called the domination number of the graph G (dominationnumber), referred to as the gamma (G). Independent set is defined as a collection of S (?) V (G), does not contain two adjacent vertices of the set S. For any point set, if its true sub-set is S, then it is not independent, then S is called a maximal independent set of the graph G (Independent set). If any of the graph G is a maximal independent set S ~~ (maximal independent set) meet | S ~~ * | ≤ | S | maximal independent set S is called a graph G is a maximal independent set (minimum independent set) . The number of elements contained in the largest independent set S is called the independence number of a graph G (independence number), denoted by α (G). In this paper, computer independent of the number of cycle diagram obtained relatively small n and k generalized Petersen graph domination number, and construct the corresponding independent set and dominating set, and to find out the law, the introduction of large n and k independent sets and dominating set, to determine the cycle diagram C (n; {1, k}), n = 3k, 4k domination number supremum and generalized Petersen graph P (n, k), k = 1, 2, 3, the number of independent infimum. This paper the results obtained and strictly proved the the cyclic graph C (n; {k}), n = 3k, 4k the number under the domination supremum and generalized Petersen graphs P (n, k), k = 1 , 2,3,5 a number of independent supremum, to arrive at the exact value of the two.
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