## The domination numbers of the 5$${\times} n$$ and 6$${\times} n$$ grid graphs.(English)Zbl 0780.05030

The $$k\times n$$ grid graph is the Cartesian product $$P_ k\times P_ n$$, where $$P_ k$$, $$P_ n$$ denote the paths of lengths $$k-1$$, $$n-1$$ respectively. A dominating set in a graph $$G$$ is a subset $$D$$ of the vertex set $$V(G)$$ of $$G$$ such that for each $$x\in V(G)-D$$ there exists a vertex $$y\in D$$ adjacent to $$x$$. The domination number of a graph is the minimum number of vertices of a dominating set in that graph. The domination number of the $$k\times n$$ grid graph is denoted by $$\gamma_{k,n}$$. E. O. Hare has developed an algorithm for computing it. She conjectured that certain formulae hold for $$\gamma_{5,n}$$ and $$\gamma_{6,n}$$ for all $$n$$ and verified this for $$n\leq 500$$. The authors of the present paper prove these formulae for all $$n$$.

### MSC:

 05C35 Extremal problems in graph theory 05C99 Graph theory

### Keywords:

grid graph; dominating set; domination number
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### References:

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