In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set \mathbb Z^2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set \mathbb Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on \mathbb Z^2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane \mathbb Z^2 for the study of digital images.

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Šlapal, Josef