On the number of shortest paths in Cartesian product graphs and its robustness

In this paper, we establish the maximum number of basic shortest paths in Cartesian product graphs and bounds on the maximum number of the vertex-disjoint shortest paths and on this of the edgedisjoint shortest paths. To the best of our knowledge, the class of Cartesian product graphs has been intensively studied according to various invariants, except the maximum number of (basic, vertex-disjoint or edge-disjoint) shortest paths, whereas the latter invariants were investigated for other graph classes. The main contribution of this paper is to fill this gap. Moreover, we investigate the impact of a vertex or an edge removal on the maximum number of basic shortest paths in Cartesian product graphs.

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