Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G   be a graph, and let ρ∈[0,1]ρ∈[0,1]. For a set D of vertices of G  , let the set Hρ(D)Hρ(D) arise by starting with the set D, and iteratively adding further vertices u   to the current set if they have at least ⌈ρdG(u)⌉⌈ρdG(u)⌉ neighbors in it. If Hρ(D)Hρ(D) contains all vertices of G, then D   is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function u↦⌈ρdG(u)⌉u↦⌈ρdG(u)⌉. Let hρ(G)hρ(G) be the minimum cardinality of such an irreversible dynamic monopoly.#R##N##R##N#For a connected graph G   of maximum degree at least 1ρ, Chang shod hρ(G)≤5.83ρn(G)hρ(G)≤5.83ρn(G), which was improved by Chang and Lyuu to hρ(G)≤4.92ρn(G)hρ(G)≤4.92ρn(G).  show that for every ϵ>0ϵ>0, there is some ρ(ϵ)∈(0,1)ρ(ϵ)∈(0,1) such that hρ(G)≤(2+ϵ)ρn(G)hρ(G)≤(2+ϵ)ρn(G) for every ρ   in (0,ρ(ϵ))(0,ρ(ϵ)), and every connected graph G   that has maximum degree at least 1ρ and girth at least 5. Furthermore,  show that hρ(T)≤ρn(T)hρ(T)≤ρn(T) for every ρ   in (0,1](0,1], and every tree T   that has order at least 1ρ.
['Michael Gentner', 'Dieter Rautenbach']
I am writing a research paper on a idea Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees
[{"from": "human", "value": "I am writing a research paper on a idea Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G \u00a0 be a graph, and let \u03c1\u2208[0,1]\u03c1\u2208[0,1]. For a set D of vertices of G \u00a0, let the set H\u03c1(D)H\u03c1(D) arise by starting with the set D, and iteratively adding further vertices u \u00a0 to the current set if they have at least \u2308\u03c1dG(u)\u2309\u2308\u03c1dG(u)\u2309 neighbors in it. If H\u03c1(D)H\u03c1(D) contains all vertices of G, then D \u00a0 is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function u\u21a6\u2308\u03c1dG(u)\u2309u\u21a6\u2308\u03c1dG(u)\u2309. Let h\u03c1(G)h\u03c1(G) be the minimum cardinality of such an irreversible dynamic monopoly.#R##N##R##N#For a connected graph G \u00a0 of maximum degree at least 1\u03c1, Chang shod h\u03c1(G)\u22645.83\u03c1n(G)h\u03c1(G)\u22645.83\u03c1n(G), which was improved by Chang and Lyuu to h\u03c1(G)\u22644.92\u03c1n(G)h\u03c1(G)\u22644.92\u03c1n(G).  show that for every \u03f5>0\u03f5>0, there is some \u03c1(\u03f5)\u2208(0,1)\u03c1(\u03f5)\u2208(0,1) such that h\u03c1(G)\u2264(2+\u03f5)\u03c1n(G)h\u03c1(G)\u2264(2+\u03f5)\u03c1n(G) for every \u03c1 \u00a0 in (0,\u03c1(\u03f5))(0,\u03c1(\u03f5)), and every connected graph G \u00a0 that has maximum degree at least 1\u03c1 and girth at least 5. Furthermore,  show that h\u03c1(T)\u2264\u03c1n(T)h\u03c1(T)\u2264\u03c1n(T) for every \u03c1 \u00a0 in (0,1](0,1], and every tree T \u00a0 that has order at least 1\u03c1."}]