Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 ≤ ki ≤ di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ VsDk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) ∣ pi ∈ R, 0 ≤ pi ≤ 1, i = 1, …, n}. An O(Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with ∣Dk∣≤fk(p).
['Jochen Harant', 'Anja Pruchnewski', 'Margit Voigt']
I am writing a research paper on a idea On Dominating Sets and Independent Sets of Graphs
[{"from": "human", "value": "I am writing a research paper on a idea On Dominating Sets and Independent Sets of Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For a graph G on vertex set V = {1, \u2026, n} let k = (k1, \u2026, kn) be an integral vector such that 1 \u2264 ki \u2264 di for i \u2208 V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk \u2286 V such that every vertex i \u2208 VsDk has at least ki neighbours in Dk. The k-domination number \u03b3k(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = \u00b7 \u00b7 \u00b7 = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, \u2026, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that \u03b3k(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, \u2026, pn) \u2223 pi \u2208 R, 0 \u2264 pi \u2264 1, i = 1, \u2026, n}. An O(\u039422\u0394n-algorithm is presented, where \u0394 is the maximum degree of G, with INPUT: p \u2208 Cn and OUTPUT: a k-dominating set Dk of G with \u2223Dk\u2223\u2264fk(p)."}]