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).

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 $A^n$.

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 Chromatic Number for a Generalization of Cartesian Product Graphs

10.37236/160 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel Král' ◽  
Douglas B. West
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Rectangular Grid ◽  
Product Graphs ◽  
Cartesian Product Graphs

Let ${\cal G}$ be a class of graphs. A $d$-fold grid over ${\cal G}$ is a graph obtained from a $d$-dimensional rectangular grid of vertices by placing a graph from ${\cal G}$ on each of the lines parallel to one of the axes. Thus each vertex belongs to $d$ of these subgraphs. The class of $d$-fold grids over ${\cal G}$ is denoted by ${\cal G}^d$. Let $f({\cal G};d)=\max_{G\in{\cal G}^d}\chi(G)$. If each graph in ${\cal G}$ is $k$-colorable, then $f({\cal G};d)\le k^d$. We show that this bound is best possible by proving that $f({\cal G};d)=k^d$ when ${\cal G}$ is the class of all $k$-colorable graphs. We also show that $f({\cal G};d)\ge{\left\lfloor\sqrt{{d\over 6\log d}}\right\rfloor}$ when ${\cal G}$ is the class of graphs with at most one edge, and $f({\cal G};d)\ge {\left\lfloor{d\over 6\log d}\right\rfloor}$ when ${\cal G}$ is the class of graphs with maximum degree $1$.

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