Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can occur. It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals, such that every clause contains exactly 2 true literals and 2 false literals is $\mathbf{NP}$-complete. In this work,  show that deciding the satisfiability of 3-SAT with the property that every literal appears in exactly 4 clauses (2 time negated and 2 times not negated), is $ \mathbf{NP} $-complete.  call this problem $(2/2/3)$-SAT. For a nonempty regular graph $G = (V,E)$, it was asked by \cite{dehhh} to determine whether for a given independent set $T $, there is an independent dominating set $D$ for $T$ such that $ T \cap D =\varnothing $? As an application of $(2/2/3)$-SAT problem,  show that for every $r\geq 3$, this problem is $ \mathbf{NP} $-complete. It is ll-known that the vertex set of every graph without isolated vertices can be partitioned into two dominating sets \cite{ID3}. Determining the computational complexity of deciding whether the vertices of a given connected cubic graph $G$ can be partitioned into independent dominating sets remains unsolved.  show that this problem is $ \mathbf{NP} $-complete, even if restricted to (i) connected graphs with only two numbers in their degree sets, (ii) cubic graphs. Finally,  study the relationship beten 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results,  prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $ \mathbf{NP} $-complete.
['Arash Ahadi', 'Ali Dehghan']
I am writing a research paper on a idea (2/2/3)-SAT problem and its applications in dominating set problems
[{"from": "human", "value": "I am writing a research paper on a idea (2/2/3)-SAT problem and its applications in dominating set problems"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The satisfiability problem is known to be $\\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can occur. It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals, such that every clause contains exactly 2 true literals and 2 false literals is $\\mathbf{NP}$-complete. In this work,  show that deciding the satisfiability of 3-SAT with the property that every literal appears in exactly 4 clauses (2 time negated and 2 times not negated), is $ \\mathbf{NP} $-complete.  call this problem $(2/2/3)$-SAT. For a nonempty regular graph $G = (V,E)$, it was asked by \\cite{dehhh} to determine whether for a given independent set $T $, there is an independent dominating set $D$ for $T$ such that $ T \\cap D =\\varnothing $? As an application of $(2/2/3)$-SAT problem,  show that for every $r\\geq 3$, this problem is $ \\mathbf{NP} $-complete. It is ll-known that the vertex set of every graph without isolated vertices can be partitioned into two dominating sets \\cite{ID3}. Determining the computational complexity of deciding whether the vertices of a given connected cubic graph $G$ can be partitioned into independent dominating sets remains unsolved.  show that this problem is $ \\mathbf{NP} $-complete, even if restricted to (i) connected graphs with only two numbers in their degree sets, (ii) cubic graphs. Finally,  study the relationship beten 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results,  prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $ \\mathbf{NP} $-complete."}]