An improved upper bound on the double Roman domination number of graphs with minimum degree at least two

An outer-independent double Roman dominating function (OIDRDF) of a graph G is a function h:V(G)→{0,1,2,3} such that i) every vertex v with f(v)=0 is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex v with f(v)=1 is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination number γoidR (G) is the minimum weight of an OIDRDF on G. It has been shown that for any tree T of order n ≥ 3, γoidR (T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, γoidR (T) ≤ 4n/3.

Download Full-text

Double Roman Graphs in P(3k, k)

Mathematics ◽

10.3390/math9040336 ◽

2021 ◽

Vol 9 (4) ◽

pp. 336

Author(s):

Zehui Shao ◽

Rija Erveš ◽

Huiqin Jiang ◽

Aljoša Peperko ◽

Pu Wu ◽

...

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Roman Domination ◽

Generalized Petersen Graphs ◽

Roman Domination Number ◽

Double Roman Domination ◽

Petersen Graphs ◽

Roman Dominating Function ◽

Sharp Lower Bound ◽

Dominating Function

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

Download Full-text

Critical concept for double Roman domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500202 ◽

2020 ◽

Vol 12 (02) ◽

pp. 2050020

Author(s):

S. Nazari-Moghaddam ◽

L. Volkmann

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Roman Domination ◽

Critical Graphs ◽

Roman Domination Number ◽

Double Roman Domination ◽

Number Formula ◽

Roman Dominating Function ◽

Domination In Graphs ◽

Dominating Function

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices assigned a [Formula: see text] or to at least one vertex assigned a [Formula: see text] and (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] The weight of a DRDF is the sum of its function values over all vertices. The double Roman domination number [Formula: see text] equals the minimum weight of a DRDF on [Formula: see text] The concept of criticality with respect to various operations on graphs has been studied for several domination parameters. In this paper, we study the concept of criticality for double Roman domination in graphs. In addition, we characterize double Roman domination edge super critical graphs and we will give several characterizations for double Roman domination vertex (edge) critical graphs.

Download Full-text