The Double Roman Domination Number of Sunlet Graphs S n

Source abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V ( G ) to {0, 1, 2, 3} in such a way that a vertex u for which f ( u ) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f ( u ) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w ( f ) = ∑ u ∈ V ( G ) f ( u ). The minimum weight of a DRDF on a graph G is called the double Roman domination number γ dR ( G ) of G . A Sunlet graph S n is constructed from an n -cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs S n for integer n ≥3. Using three lemmas, we derive the exact formula γ dR ( S n )=⌈7 n /3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ ( S n )= n with the known inequality 2 γ ( G )≤ γ dR ( G )≤3 γ ( G ) for double Roman graphs, we prove that S n fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.