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.

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 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.

Vertex criticality with respect to isolate domination

2015 ◽  
Vol 07 (02) ◽  
pp. 1550010 ◽  
Author(s):  
I. Sahul Hamid ◽  
S. Balamurugan
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Extended Chain ◽  
Isolated Vertex ◽  
Domination In Graphs

A set S of vertices of a graph G is called a dominating set of G if every vertex in V(G) - S is adjacent to a vertex in S. A dominating set S such that the subgraph 〈S〉 induced by S has at least one isolated vertex is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate domination number and is denoted by γ0(G). This concept was introduced in [I. Sahul Hamid and S. Balamurugan, Isolate domination in graphs (Communicated)] and further studied in [I. Sahul Hamid and S. Balamurugan, Extended chain of domination parameters in graphs, ISRN Combin. 2013 (2013), Article ID: 792743, 4 pp.; Isolate domination and maximum degree, Bull. Int. Math. Virtual Inst. 3 (2013) 127–133; Isolate domination in unicyclic graphs, Int. J. Math. Soft Comput. 3(3) (2013) 79–83]. This paper studies the effect of the removal of a vertex upon the isolate domination number.

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.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

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.

scholarly journals Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture

10.37236/5147 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Florent Foucaud ◽  
Michael A. Henning
Keyword(s):  
Dominating Set ◽  
Open Neighborhood ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods.  The location-total domination number of $G$, denoted $\gamma_t^L(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \ge 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $\gamma_t^L(G) \le \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $\gamma_t^L(G) \le \frac{3}{4}n$.

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 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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