Bounds for signed double Roman <i>k</i>-domination in trees

Abstract

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V ( G ). A signed double Roman k-dominating function (SDR k DF) on a graph G is a function f : V ( G ) → {−1,1,2,3} such that (i) every vertex v with f ( v ) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f ( w ) = 3, (ii) every vertex v with f ( v ) = 1 is adjacent to at least one vertex w with f ( w ) ≥ 2 and (iii) ∑ u ∈ N [ v ] f ( u ) ≥ k holds for any vertex v . The weight of a SDR k DF f is ∑ u ∈ V ( G ) f ( u ), and the minimum weight of a SDR k DF is the signed double Roman k-domination number γ k sdR ( G ) of G . In this paper, we investigate the signed double Roman k -domination number of trees. In particular, we present lower and upper bounds on γ k sdR ( T ) for 2 ≤ k ≤ 6 and classify all extremal trees.