Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

2021 ◽  
pp. 2150090
Author(s):  
J. Maria Regila Baby ◽  
K. Uma Samundesvari
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Cartesian Product Graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

scholarly journals Cartesian product graphs and k-tuple total domination

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6713-6731
Author(s):  
Adel Kazemi ◽  
Behnaz Pahlavsay ◽  
Rebecca Stones
Keyword(s):  
Dominating Set ◽  
Cartesian Product ◽  
Minimum Size ◽  
Constructive Proof ◽  
Extremal Case ◽  
Packing Number ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Open Packing ◽  
Cartesian Product Graphs

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted ?xk,t(G). We give a Vizing-like inequality for Cartesian product graphs, namely ?xk,t(G) ?xk,t(H)? 2k?xk,t(G_H) provided ?xk,t(G) ? 2k?(G) holds, where ? denotes the packing number. We also give bounds on ?xk,t(G_H) in terms of (open) packing numbers, and consider the extremal case of ?xk,t(Kn_Km), i.e., the rook?s graph, giving a constructive proof of a general formula for ?x2,t(Kn_Km).

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.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

scholarly journals New Bounds on the Double Total Domination Number of Graphs

2021 ◽  
Author(s):  
A. Cabrera-Martínez ◽  
F. A. Hernández-Mira
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

scholarly journals On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

On the fixed-point type Sylvester matrix equations over complete commutative dioids

2014 ◽  
Vol 27 ◽  
Author(s):  
Benham Hashemi ◽  
Mahtab Mirzaei Khalilabadi ◽  
Hanieh Tavakolipour
Keyword(s):  
Fixed Point ◽  
Tensor Product ◽  
Cartesian Product ◽  
Matrix Equations ◽  
Minimum Cardinality ◽  
Sylvester Matrix ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Sylvester Matrix Equations ◽  
Point Type

This paper extends the concept of tropical tensor product defined by Butkovic and Fiedler to general idempotent dioids. Then, it proposes an algorithm in order to solve the fixed-point type Sylvester matrix equations of the form X = A ⊗ X ⊕ X ⊗ B ⊕ C. An application is discussed in efficiently solving the minimum cardinality path problem in Cartesian product graphs.

scholarly journals Total Domination and Matching Numbers in Claw-Free Graphs

10.37236/1085 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Free Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $M$ of edges of a graph $G$ is a matching if no two edges in $M$ are incident to the same vertex. The matching number of $G$ is the maximum cardinality of a matching of $G$. A set $S$ of vertices in $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.

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