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locating-dominating set; location-domatic partition; location-domatic number; domatic number
A subset $D$ of the vertex set $V(G)$ of a graph $G$ is called locating-dominating, if for each $x\in V(G)-D$ there exists a vertex $y\to D$ adjacent to $x$ and for any two distinct vertices $x_1$, $x_2$ of $V(G)-D$ the intersections of $D$ with the neighbourhoods of $x_1$ and $x_2$ are distinct. The maximum number of classes of a partition of $V(G)$ whose classes are locating-dominating sets in $G$ is called the location-domatic number of $G.$ Its basic properties are studied.
[1] E. J. Cockayne S. T. Hedetniemi: Towards a theory of domination in graphs. Networks 7 (1977), 247-261. DOI 10.1002/net.3230070305 | MR 0483788
[2] D. F. Rall P. J. Slater: On location-domination numbers for certain classes of graphs. Congressus Numerantium 45 (1984), 77-106. MR 0777715
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