(2/2/3)-SAT problem and its applications in dominating set problems

The satisfiability problem is known to be 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 NP-complete. In this work, we 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 NP-complete. We call this problem (2/2/3)-SAT. For a nonempty regular graph G = (V,E), it was asked by [1] to determine whether for a given independent set T , there is an independent dominating set D for T such that T ∩ D = ∅? As an application of (2/2/3)-SAT problem, we show that for every r ≥ 3, this problem is NP-complete. It is well-known that the vertex set of every graph without isolated vertices can be partitioned into two dominating sets [10]. 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. We show that this problem is NP-complete, even if restricted to (i) connected graphs with only two numbers in their degree sets, (ii) cubic graphs. Finally, we study the relationship between 1perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph G deciding whether G is 4-incidence colorable is NP-complete. ∗E-mail addresses: arash [email protected] (Arash Ahadi), ali [email protected], [email protected] (Ali Dehghan).