Lights Out!: A Survey of Parity Domination in Grid Graphs

On independent domination numbers of grid and toroidal grid directed graphs

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

On Generalizing the “Lights Out” Game and a Generalization of Parity Domination

The Lights Out game on a graph G is played as follows. Begin with a (not necessarily proper) coloring of V (G) with elements of Z2. When a vertex is toggled, that vertex and all adjacent vertices change their colors from 0 to 1 or vice-versa. The game is won when all vertices have color 0. The winnability of this game is related to the existence of a parity dominating set. We generalize this ga...

Generalized switch-setting problems

Switch-setting games like Lights Out are typically modelled as a graph, where the vertices represent switches and lamps, and the edges capture the switching rules. We generalize the concepts used for a mathematical description of Lights Out and its relatives. Our approach uses bipartite graphs and allows for the analysis of a broader class of switch-setting games, which is demonstrated at a new...

Fibonacci Polynomials and Parity Domination in Grid Graphs

Domination and Signed Domination Number of Cayley Graphs

In this paper, we investigate domination number as well as signed domination numbers of Cay(G : S) for all cyclic group G of order n, where n in {p^m; pq} and S = { a^i : i in B(1; n)}. We also introduce some families of connected regular graphs gamma such that gamma_S(Gamma) in {2,3,4,5 }.