Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let G=(V,E)G=(V,E) be a graph with no isolated vertex. A subset of vertices SS is a total dominating set if every vertex of GG is adjacent to some vertex of SS. For some αα with 0<α≤10<α≤1, a total dominating set SS in GG is an αα-total dominating set if for every vertex v∈V∖Sv∈V∖S, |N(v)⋂S|⩾α|N(v)||N(v)⋂S|⩾α|N(v)|. The αα-total domination number of GG, denoted by γαt(G)γαt(G), is the minimum cardinality of an αα-total dominating set of GG. In Henning and Rad (2012), Henning and Rad posed the following question: Let GG be a connected cubic graph with order nn. Is it true that γαt(G)≤3n4 for 23<α≤1 ?#R##N##R##N#,  give a positive ansr toward this question. Furthermore,  give a characterization on cubic graphs attaining the bound for the αα-total domination number.
['Xue-gang Chen', 'Ting Gao']
I am writing a research paper on a idea A note on α -total domination in cubic graphs
[{"from": "human", "value": "I am writing a research paper on a idea A note on \u03b1 -total domination in cubic graphs"}, {"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=(V,E)G=(V,E) be a graph with no isolated vertex. A subset of vertices SS is a total dominating set if every vertex of GG is adjacent to some vertex of SS. For some \u03b1\u03b1 with 0<\u03b1\u226410<\u03b1\u22641, a total dominating set SS in GG is an \u03b1\u03b1-total dominating set if for every vertex v\u2208V\u2216Sv\u2208V\u2216S, |N(v)\u22c2S|\u2a7e\u03b1|N(v)||N(v)\u22c2S|\u2a7e\u03b1|N(v)|. The \u03b1\u03b1-total domination number of GG, denoted by \u03b3\u03b1t(G)\u03b3\u03b1t(G), is the minimum cardinality of an \u03b1\u03b1-total dominating set of GG. In Henning and Rad (2012), Henning and Rad posed the following question: Let GG be a connected cubic graph with order nn. Is it true that \u03b3\u03b1t(G)\u22643n4 for 23<\u03b1\u22641 ?#R##N##R##N#,  give a positive ansr toward this question. Furthermore,  give a characterization on cubic graphs attaining the bound for the \u03b1\u03b1-total domination number."}]