5 research outputs found
-SAT problem and its applications in dominating set problems
The satisfiability problem is known to be -complete in general
and for many restricted cases. One way to restrict instances of -SAT is to
limit the number of times a variable can be occurred. It was shown that for an
instance of 4-SAT with the property that every variable appears in exactly 4
clauses (2 times negated and 2 times not negated), determining whether there is
an assignment for variables such that every clause contains exactly two true
variables and two false variables is -complete. In this work, we
show that deciding the satisfiability of 3-SAT with the property that every
variable appears in exactly four clauses (two times negated and two times not
negated), and each clause contains at least two distinct variables is -complete. We call this problem -SAT. For an -regular
graph with , it was asked in [Discrete Appl. Math.,
160(15):2142--2146, 2012] to determine whether for a given independent set
there is an independent dominating set that dominates such that ? As an application of -SAT problem we show that
for every , this problem is -complete. Among other
results, we study the relationship between 1-perfect codes and the incidence
coloring of graphs and as another application of our complexity results, we
prove that for a given cubic graph deciding whether is 4-incidence
colorable is -complete
Interval Incidence Coloring of Subcubic Graphs
In this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is NP-complete, and they asked if Xii(G) ≤ 2Δ(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with Δ(G) = 3
Interval Incidence Coloring of Subcubic Graphs
In this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is NP-complete, and they asked if Xii(G) ≤ 2Δ(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with Δ(G) = 3