Co-Roman domination in graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

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Resolving Roman domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500288 ◽

2021 ◽

Author(s):

P. Roushini Leely Pushpam ◽

B. Mahavir ◽

M. Kamalam

Keyword(s):

Minimum Weight ◽

Domination Number ◽

Roman Domination ◽

Roman Domination Number ◽

Roman Dominating Function ◽

Domination In Graphs ◽

Dominating Function

Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.

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Algorithmic Aspects of Total Roman and Total Double Roman Domination in Graphs

Algorithms and Discrete Applied Mathematics - Lecture Notes in Computer Science ◽

10.1007/978-3-030-67899-9_3 ◽

2021 ◽

pp. 32-42

Author(s):

Chakradhar Padamutham ◽

Venkata Subba Reddy Palagiri

Keyword(s):

Roman Domination ◽

Double Roman Domination ◽

Domination In Graphs

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Global double Roman domination in graphs

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2019.1569833 ◽

2019 ◽

Vol 22 (1) ◽

pp. 31-44 ◽

Cited By ~ 4

Author(s):

Zehui Shao ◽

S. M. Sheikholeslami ◽

S. Nazari-Moghaddam ◽

Shaohui Wang

Keyword(s):

Roman Domination ◽

Double Roman Domination ◽

Domination In Graphs

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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.

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Edge lifting and Roman domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500646 ◽

2020 ◽

pp. 2150064

Author(s):

Hicham Meraimi ◽

Mustapha Chellali

Keyword(s):

Domination Number ◽

Roman Domination ◽

Roman Domination Number ◽

Induced Path ◽

Domination In Graphs

Let [Formula: see text] be a graph, and let [Formula: see text] be an induced path centered at [Formula: see text]. An edge lift defined on [Formula: see text] is the action of removing edges [Formula: see text] and [Formula: see text] while adding the edge [Formula: see text] to the edge set of [Formula: see text]. In this paper, we initiate the study of the effects of edge lifting on the Roman domination number of a graph, where various properties are established. A characterization of all trees for which every edge lift increases the Roman domination number is provided. Moreover, we characterize the edge lift of a graph decreasing the Roman domination number, and we show that there are no graphs with at most one cycle for which every possible edge lift can have this property.

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Majority Roman domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500622 ◽

2020 ◽

pp. 2150062

Author(s):

S. Anandha Prabhavathy

Keyword(s):

Domination Number ◽

Roman Domination ◽

Roman Domination Number ◽

Roman Dominating Function ◽

Closed Neighborhood ◽

Domination In Graphs ◽

Dominating Function

A Majority Roman Dominating Function (MRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) the sum of its function values over at least half the closed neighborhood is at least one and (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a MRDF is the sum of its function values over all vertices. The Majority Roman Domination Number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we initiate the study of Majority Roman Domination in Graphs.

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