Research Article

Bounds for signed double Roman k-domination in trees

¹ School of Information Science and Engineering Chengdu University, 610106 Chengdu, China
² Institute of Computing Science and Technology Guangzhou University, 510006 Guangzhou, China
³ Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
⁴ Center for Cyber Security University of Electronic Science and Technology of China 611731 Chengdu, China

^* Corresponding author: s.m.sheikholeslami@azaruniv.ac.ir

Received: 29 September 2017
Accepted: 30 May 2018

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 (SDRkDF) 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 SDRkDF f is ∑_u∈V(G) f(u), and the minimum weight of a SDRkDF 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.