AUTHOR=Rao Yongsheng , Kosari Saeed , Sheikholeslami Seyed Mahmoud , Chellali M. , Kheibari Mahla 

TITLE=On the Outer-Independent Double Roman Domination of Graphs

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=6

YEAR=2021

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2020.559132

DOI=10.3389/fams.2020.559132

ISSN=2297-4687

ABSTRACT=<p>An outer-independent double Roman dominating function (OIDRDF) of a graph <italic>G</italic> is a function <inline-formula id="inf1"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>→</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0,1,2,3</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> such that i) every vertex <italic>v</italic> with <inline-formula id="inf2"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex <italic>v</italic> with <inline-formula id="inf3"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> 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 <italic>γ</italic><sub><italic>oidR</italic></sub> (<italic>G</italic>) is the minimum weight of an OIDRDF on <italic>G</italic>. It has been shown that for any tree <italic>T</italic> of order <italic>n</italic> ≥ 3, <italic>γ</italic><sub><italic>oidR</italic></sub> (<italic>T</italic>) ≤ 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 <italic>G</italic> of order <italic>n</italic> with minimum degree at least two in which the set of vertices with degree at least three is independent, <italic>γ</italic><sub><italic>oidR</italic></sub> (<italic>T</italic>) ≤ 4n/3.</p>