LIGHTS OUT! on Cartesian Products

2017 ◽  
Vol 32 ◽  
pp. 464-474
Author(s):  
Travis Peters ◽  
John Goldwasser ◽  
Michael Young
Keyword(s):  
Open Neighborhood ◽  
Cartesian Product ◽  
Finite Sequence ◽  
Initial Configuration ◽  
Cartesian Products ◽  
Simple Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Arbitrary Graphs ◽  
Universally Solvable

The game LIGHTS OUT! is played on a 5 by 5 square grid of buttons; each button may be on or off. Pressing a button changes the on/o state of the light of the button pressed and of all its vertical and horizontal neighbors. Given an initial configuration of buttons that are on, the object of the game is to turn all the lights out. The game can be generalized to arbitrary graphs. In this paper, Cartesian Product graphs (that is, graphs of the form G\box H, where G and H are arbitrary finite, simple graphs) are investigated. In particular, conditions for which GH is universally solvable (every initial configuration of lights can be turned out by a finite sequence of button presses), using both closed neighborhood switching and open neighborhood switching, are provided.

scholarly journals Connected domination game played on Cartesian products

2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Arbitrary Graph ◽  
Cartesian Products ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Game Domination Number ◽  
Domination Game

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

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 $1$.

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 $A^n$.

Total k-domination in Cartesian product graphs

2017 ◽  
Vol 75 (2) ◽  
pp. 255-267 ◽  
Author(s):  
S. Bermudo ◽  
J. L. Sanchéz ◽  
J. M. Sigarreta
Keyword(s):  
Cartesian Product ◽  
Product Graphs ◽  
Cartesian Product Graphs
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