scholarly journals New results on connected dominating structures in graphs

2019 ◽  
Vol 11 (1) ◽  
pp. 52-64
Author(s):  
Libin Chacko Samuel ◽  
Mayamma Joseph
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Forbidden Subgraph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Complete Subgraph ◽  
Strong Product ◽  
Necessary And Sufficient ◽  
Domination In Graphs ◽  
Product Of Graphs

Abstract A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found.

scholarly journals On k-Fair Total Domination in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 578-589
Author(s):  
Wardah Masanggila Bent-Usman ◽  
Rowena T. Isla
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination In Graphs ◽  
Product Of Graphs

Let G = (V (G), E(G)) be a simple non-empty graph. For an integer k ≥ 1, a k-fairtotal dominating set (kf td-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkf td(G), is the minimum cardinality of a kf td-set. A k-fair total dominating set of cardinality γkf td(G) is called a minimum k-fair total dominating set or a γkf td-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharpbounds of their corresponding k-fair total domination number.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

scholarly journals Resolving Restrained Domination in Graphs

2021 ◽  
Vol 14 (3) ◽  
pp. 829-841
Author(s):  
Gerald Bacon Monsanto ◽  
Helen M. Rara
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Restrained Domination ◽  
Isolated Vertex ◽  
Domination In Graphs ◽  
Product Of Graphs ◽  
Restrained Domination Number

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

Doubly connected bi-domination in graphs

2020 ◽  
pp. 2150009 ◽  
Author(s):  
Mohammed A. Abdlhusein
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Finite Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Isolated Vertex ◽  
Domination In Graphs

Let [Formula: see text] be a finite graph, simple, undirected and has no isolated vertex. A dominating subset [Formula: see text] of [Formula: see text] is said a bi-dominating set, if every vertex of it dominates two vertices of [Formula: see text]. The bi-domination number of [Formula: see text], denoted by [Formula: see text] is the minimum cardinality over all bi-dominating sets in [Formula: see text]. In this paper, a certain modified bi-domination parameter called doubly connected bi-domination and its inverse are introduced. Several bounds and properties are studied here. These modified dominations are applied and evaluated for several well-known graphs and complement graphs.

Geodetic global domination in corona and strong product of graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050043
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Strong Product ◽  
Geodetic Set ◽  
Product Of Graphs ◽  
Global Domination Number

A set S of vertices in a connected graph [Formula: see text] is called a geodetic set if every vertex not in [Formula: see text] lies on a shortest path between two vertices from [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] has at least one neighbor in [Formula: see text]. A set [Formula: see text] is called a geodetic global dominating set of [Formula: see text] if [Formula: see text] is both geodetic and global dominating set of [Formula: see text]. The geodetic global domination number is the minimum cardinality of a geodetic global dominating set in [Formula: see text]. In this paper, we determine the geodetic global domination number of the corona and strong products of two graphs.

Connected hop domination in graphs under some binary operations

2018 ◽  
Vol 11 (05) ◽  
pp. 1850075
Author(s):  
Yamilita M. Pabilona ◽  
Helen M. Rara
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Induced Subgraph ◽  
Binary Operations ◽  
Network Application ◽  
Domination In Graphs ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph. A hop dominating set [Formula: see text] is called a connected hop dominating set of [Formula: see text] if the induced subgraph [Formula: see text] of [Formula: see text] is connected. The smallest cardinality of a connected hop dominating set of [Formula: see text], denoted by [Formula: see text], is called the connected hop domination number of [Formula: see text]. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

Outer-convex domination in graphs

2019 ◽  
Vol 12 (01) ◽  
pp. 2050008 ◽  
Author(s):  
Jonecis A. Dayap ◽  
Enrico L. Enriquez
Keyword(s):  
Convex Set ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Convex Domination Number ◽  
Domination In Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] of vertices of a graph [Formula: see text] is an outer-convex dominating set if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text] and [Formula: see text] is a convex set. The outer-convex domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-convex dominating set of [Formula: see text]. An outer-convex dominating set of cardinality [Formula: see text] will be called a [Formula: see text]-[Formula: see text]. In this paper, we initiate the study and characterize the outer-convex dominating sets in the join of the two graphs.

Movable resolving domination in graphs

2021 ◽  
Author(s):  
Gerald B. Monsanto ◽  
Helen M. Rara
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Domination In Graphs

Let [Formula: see text] be a connected graph. Brigham et al., Resolving domination in graphs, Math. Bohem. 1 (2003) 25–36 defined a resolving dominating set as a set [Formula: see text] of vertices of a connected graph [Formula: see text] that is both resolving and dominating. A resolving dominating is a [Formula: see text]-movable resolving dominating set of [Formula: see text] if for every [Formula: see text], either [Formula: see text] is a resolving dominating set or there exists a vertex [Formula: see text] such that [Formula: see text] is a resolving dominating set of [Formula: see text]. The minimum cardinality of a [Formula: see text]-movable resolving dominating set of [Formula: see text], denoted by [Formula: see text] is the [Formula: see text]-movable[Formula: see text]-domination number of [Formula: see text]. A [Formula: see text]-movable resolving dominating set with cardinality [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we characterize the [Formula: see text]-movable resolving dominating sets in the join and lexicographic product of two graphs and determine the bounds or exact values of the [Formula: see text]-movable resolving domination number of these graphs.

scholarly journals Total Perfect Hop Domination in Graphs Under Some Binary Operations

2021 ◽  
Vol 14 (3) ◽  
pp. 803-815
Author(s):  
Raicah Cayongcat Rakim ◽  
Helen M Rara
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Binary Operations ◽  
Domination In Graphs ◽  
Product Of Graphs

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ V (G) is a perfect hop dominating set of G if for every v ∈ V (G) \ S, there is exactly one vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of a perfect hop dominating set of G is called the perfect hop domination number of G, denoted by γph(G). A perfect hop dominating set S ⊆ V (G) is called a total perfect hop dominating set of G if for every v ∈ V (G), there is exactly one vertex u ∈ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by γtph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality γtph(G) is referred to as a γtph-set of G. In this paper, we characterize the total perfect hop dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding total perfect hop domination number.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

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