Observability of Lattice Graphs

We consider a graph observability problem: how many edge colors are needed for an unlabeled graph so that an agent, walking from node to node, can uniquely determine its location from just the observed color sequence of the walk? Specifically, let G(n, d) be an edge-colored subgraph of d-dimensional (directed or undirected) lattice of size $$n^d = n \times n \times \cdots \times n$$nd=n×n×⋯×n. We say that G(n, d) is t-observable if an agent can uniquely determine its current position in the graph from the color sequence of any t-dimensional walk, where the dimension is the number of different directions spanned by the edges of the walk. A walk in an undirected lattice G(n, d) has dimension between 1 and d, but a directed walk can have dimension between 1 and 2d because of two different orientations for each axis. We derive bounds on the number of colors needed for t-observability. Our main result is that $$\varTheta (n^{d/t})$$Θ(nd/t) colors are both necessary and sufficient for t-observability of G(n, d), where d is considered a constant. This shows an interesting dependence of graph observability on the ratio between the dimension of the lattice and that of the walk. In particular, the number of colors for full-dimensional walks is $$\varTheta (n^{1/2})$$Θ(n1/2) in the directed case, and $$\varTheta (n)$$Θ(n) in the undirected case, independent of the lattice dimension. All of our results extend easily to non-square lattices: given a lattice graph of size $$N = n_1 \times n_2 \times \cdots \times n_d$$N=n1×n2×⋯×nd, the number of colors for t-observability is $$\varTheta (\root t \of {N})$$Θ(Nt).