Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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.  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 ,v i} and edge set {uiui+1 ,u ivi ,v ivi+k}; where i =1 ,2 ,...,n and subscripts are reduced modulo n. First,  characterize minimum vertex covers in generalized Petersen graphs by defining a new quantity. Second,  prove that β(P(n, k)) ≥ n + (n,k)+1 2 , for all odd n, where (n, k) is the greatest common devisor of n and k; β(P(n, k)) ≤ n + k+1 2 for all odd k; for all m<k , β(P(n, k)) ≤ n m β(P(m, k (mod m))) when m | n and otherwise β(P(n, k)) ≤� n m � β(P(m, k (mod m))) + 2k. Most of these bounds are sharp. Third,  determine the exact values of β(P(n, k)) for k =1 ,3 and in the case when n is even and k is odd, also the case when both n and k are odd and k | n.  conjecture that β(P(n, k)) ≤ n + � n � , for all n and k.
['Babak Behsaz', 'Pooya Hatami', 'Ebadollah S. Mahmoodian']
I am writing a research paper on a idea On minimum vertex cover of generalized Petersen graphs
[{"from": "human", "value": "I am writing a research paper on a idea On minimum vertex cover of generalized Petersen graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Determining the size of minimum vertex cover of a graph G, denoted by \u03b2(G), is an NP-complete problem. Also, for only few families of graphs, \u03b2(G) is known.  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 ,v i} and edge set {uiui+1 ,u ivi ,v ivi+k}; where i =1 ,2 ,...,n and subscripts are reduced modulo n. First,  characterize minimum vertex covers in generalized Petersen graphs by defining a new quantity. Second,  prove that \u03b2(P(n, k)) \u2265 n + (n,k)+1 2 , for all odd n, where (n, k) is the greatest common devisor of n and k; \u03b2(P(n, k)) \u2264 n + k+1 2 for all odd k; for all m<k , \u03b2(P(n, k)) \u2264 n m \u03b2(P(m, k (mod m))) when m | n and otherwise \u03b2(P(n, k)) \u2264\ufffd n m \ufffd \u03b2(P(m, k (mod m))) + 2k. Most of these bounds are sharp. Third,  determine the exact values of \u03b2(P(n, k)) for k =1 ,3 and in the case when n is even and k is odd, also the case when both n and k are odd and k | n.  conjecture that \u03b2(P(n, k)) \u2264 n + \ufffd n \ufffd , for all n and k."}]