Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $$G=V,E$$ be a connected graph with at least two vertices. For a fixed positive integer $$b>1$$, a set $$D\subseteq V$$ is called a b-disjunctive total dominating set of G if for every vertex $$v\in V$$, v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The minimum cardinality of a b-disjunctive total dominating set of G is called the b-disjunctive total domination number of G, and is denoted by $$\gamma _{b}^{td}G$$. The Minimum b-Disj Total Domination problem is to find a b-disjunctive total dominating set of cardinality $$\gamma _{b}^{td}G$$. Given a positive integer k and a graph G, the b-Disj Total Dom Decision problem is to decide whether G has a b-disjunctive total dominating set of cardinality at most k. ,  initiate the algorithmic study of the Minimum b-Disj Total Domination problem.  prove that the b-Disj Total Dom Decision problem is NP-complete even for bipartite graphs and chordal graphs, two important graph classes. On the positive side,  propose a $$\ln \varDelta ^{2}+b-1\varDelta +1$$-approximation algorithm for the Minimum b-Disj Total Domination problem.  prove that the Minimum b-Disj Total Domination problem cannot be approximated within $$\frac{1}{2}1-\epsilon \ln |V|$$ for any $$\epsilon > 0$$ unless NP $$\subseteq $$ DTIME$$|V|^{O\log \log |V|}$$. Finally,  show that the Minimum b-Disj Total Domination problem is APX-complete for bipartite graphs with maximum degree $$b+3$$.
['Arti Pandey', 'B. Panda']
I am writing a research paper on a idea b-Disjunctive Total Domination in Graphs: Algorithm and Hardness Results
[{"from": "human", "value": "I am writing a research paper on a idea b-Disjunctive Total Domination in Graphs: Algorithm and Hardness Results"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $$G=V,E$$ be a connected graph with at least two vertices. For a fixed positive integer $$b>1$$, a set $$D\\subseteq V$$ is called a b-disjunctive total dominating set of G if for every vertex $$v\\in V$$, v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The minimum cardinality of a b-disjunctive total dominating set of G is called the b-disjunctive total domination number of G, and is denoted by $$\\gamma _{b}^{td}G$$. The Minimum b-Disj Total Domination problem is to find a b-disjunctive total dominating set of cardinality $$\\gamma _{b}^{td}G$$. Given a positive integer k and a graph G, the b-Disj Total Dom Decision problem is to decide whether G has a b-disjunctive total dominating set of cardinality at most k. ,  initiate the algorithmic study of the Minimum b-Disj Total Domination problem.  prove that the b-Disj Total Dom Decision problem is NP-complete even for bipartite graphs and chordal graphs, two important graph classes. On the positive side,  propose a $$\\ln \\varDelta ^{2}+b-1\\varDelta +1$$-approximation algorithm for the Minimum b-Disj Total Domination problem.  prove that the Minimum b-Disj Total Domination problem cannot be approximated within $$\\frac{1}{2}1-\\epsilon \\ln |V|$$ for any $$\\epsilon > 0$$ unless NP $$\\subseteq $$ DTIME$$|V|^{O\\log \\log |V|}$$. Finally,  show that the Minimum b-Disj Total Domination problem is APX-complete for bipartite graphs with maximum degree $$b+3$$."}]