Forcing Total dr-Power Domination Number of Graphs Under Some Binary Operations

In this paper, the total dr-power domination number of graphs such as complete bipartite graph, generalized fan and generalized wheel are obtained. The forcing total dr-power domination number of graphs resulting from some binary operations such as join, corona and lexicographic product of graphs were determined.

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Another H-super magic decompositions of the lexicographic product of graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.2 ◽

2018 ◽

Vol 2 (2) ◽

pp. 72

Author(s):

H Hendy ◽

Kiki A. Sugeng ◽

A.N.M Salman ◽

Nisa Ayunda

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Graph Decomposition ◽

Graph Labeling ◽

Sufficient Condition ◽

Lexicographic Product ◽

Simple Graphs ◽

Product Of Graphs

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2, denoted by G1[G2], is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn, n where n is the order of G2. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where n > 3, m > 1, and t = 3, 4, n − 2.

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Connected hop domination in graphs under some binary operations

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500754 ◽

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.

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Total Perfect Hop Domination in Graphs Under Some Binary Operations

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i3.3975 ◽

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 âŠ† V (G) is a perfect hop dominating set of G if for every v âˆˆ V (G) \ S, there is exactly one vertex u âˆˆ 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 Î³ph(G). A perfect hop dominating set S âŠ† V (G) is called a total perfect hop dominating set of G if for every v âˆˆ V (G), there is exactly one vertex u âˆˆ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by Î³tph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality Î³tph(G) is referred to as a Î³tph-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.

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Clique doubly connected domination in the join and lexicographic product of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500664 ◽

2020 ◽

Vol 12 (05) ◽

pp. 2050066

Author(s):

Enrico L. Enriquez ◽

Albert D. Ngujo

Keyword(s):

Nonempty Subset ◽

Dominating Set ◽

Domination Number ◽

Simple Graph ◽

Connected Dominating Set ◽

Lexicographic Product ◽

Connected Domination Number ◽

Vertex Set ◽

Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

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Signed domination and Mycielski’s structure in graphs

RAIRO - Operations Research ◽

10.1051/ro/2019109 ◽

2020 ◽

Vol 54 (4) ◽

pp. 1077-1086

Author(s):

Arezoo N. Ghameshlou ◽

Athena Shaminezhad ◽

Ebrahim Vatandoost ◽

Abdollah Khodkar

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Minimum Weight ◽

Domination Number ◽

Upper Bounds ◽

Underlying Graph ◽

Signed Domination ◽

Signed Domination Number ◽

Dominating Function ◽

Domination Numbers

Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

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On the power domination number of the Cartesian product of graphs

AKCE International Journal of Graphs and Combinatorics ◽

10.1016/j.akcej.2019.02.004 ◽

2019 ◽

Vol 16 (3) ◽

pp. 253-257

Author(s):

K.M. Koh ◽

K.W. Soh

Keyword(s):

Domination Number ◽

Cartesian Product ◽

Power Domination ◽

Cartesian Product Of Graphs ◽

Product Of Graphs

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Liar’s domination in graphs under some operations

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.48.2017.2188 ◽

2017 ◽

Vol 48 (1) ◽

pp. 61-71 ◽

Cited By ~ 1

Author(s):

Sergio Jr. Rosales Canoy ◽

Carlito Bancoyo Balandra

Keyword(s):

Dominating Set ◽

Domination Number ◽

Lexicographic Product ◽

Domination In Graphs ◽

Product Of Graphs

A set $S\subseteq V(G)$ is a liar's dominating set ($lds$) of graph $G$ if $|N_G[v]\cap S|\geq 2$ for every $v\in V(G)$ and $|(N_G[u]\cup N_G[v])\cap S|\geq 3$ for any two distinct vertices $u,v \in V(G)$. The liar's domination number of $G$, denoted by $\gamma_{LR}(G)$, is the smallest cardinality of a liar's dominating set of $G$. In this paper we study the concept of liar's domination in the join, corona, and lexicographic product of graphs.

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Resolving Restrained Domination in Graphs

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i3.3985 ◽

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 âŠ† 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.

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Forcing Independent Domination Number of a Graph

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v12i4.3484 ◽

2019 ◽

Vol 12 (4) ◽

pp. 1371-1381

Author(s):

Cris Laquibla Armada ◽

Sergio, Jr. R. Canoy

Keyword(s):

Domination Number ◽

Lexicographic Product ◽

Binary Operations ◽

Independent Domination

In this paper, we obtain the forcing independent domination number of some special graphs. Further, we determine the forcing independent domination number of graphs under some binary operations such join, corona and lexicographic product of two graphs.

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Dierentiating-Dominating sets in graphs Under binary operations

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.46.2015.1553 ◽

2015 ◽

Vol 46 (1) ◽

pp. 51-60

Author(s):

Sergio R.osales Canoy,Jr. ◽

Gina A. Malacas

Keyword(s):

Lexicographic Product ◽

Dominating Sets ◽

Binary Operations ◽

Product Of Graphs ◽

Domination Numbers

In this paper we characterize the differentiating-dominating sets in the join, corona, and lexicographic product of graphs. We also determine bounds or the exact differentiating-domination numbers of these graphs.

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