Double Roman domination in trees

Abstract A subset S of the vertex set of a graph G is a dominating set if every vertex of G not in S has at least one neighbor in S. The domination number γ ( G ) is defined to be the minimum cardinality among all dominating set of G. A Roman dominating function on a graph G is a function f : V ( G ) → { 0 , 1 , 2 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2 . The weight of a Roman dominating function f is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R ( G ) of G. A double Roman dominating function on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2 , and every vertex u for which f ( u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2 . The weight of a double Roman dominating function f is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number γ d R ( G ) of G. Beeler et al. (2016) [6] showed that 2 γ ( G ) ≤ γ d R ( G ) ≤ 3 γ ( G ) and showed that 2 γ ( T ) + 1 ≤ γ d R ( T ) ≤ 3 γ ( T ) for any non-trivial tree T and posed a problem that if it is possible to construct a polynomial algorithm for computing the value of γ d R ( T ) for any tree T. In this paper, we answer this problem by giving a linear time algorithm to compute the value of γ d R ( T ) for any tree T. Moreover, we give characterizations of trees with 2 γ ( T ) + 1 = γ d R ( T ) and γ d R ( T ) + 1 = 2 γ R ( T ) .