## On the domination number of the $$m\times n$$ toroidal grid graph.(English)Zbl 0976.05045

Summary: Let $$G= (V,E)$$ be a graph. A subset $$D$$ of $$V$$ is called a dominating set for $$G$$ if for every $$v\in V$$ either $$v\in D$$, or else $$v$$ is adjacent to some vertex in $$D$$. The domination number of $$G$$, denoted by $$\gamma(G)$$, is equal to the minimum cardinality of a dominating set of $$G$$. In this paper we establish the domination numbers of the toroidal graphs $$C_m\times C_n$$ of the product of two cycles $$C_m$$ and $$C_n$$ of length $$m$$ and $$n$$ for $$m= 10$$ and arbitrary $$n$$, and for $$m,n\equiv 0\pmod 5$$.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C38 Paths and cycles

### Keywords:

domination number; dominating set; toroidal graphs; cycles