On induced and isometric embeddings of graphs into the strong product of paths

The strong isometric dimension and the adjacent isometric dimension of graphs are compared. The concepts are equivalent for graphs of diameter 2 in which case the problem of determining these dimensions can be reduced to a covering problem with complete bipartite graphs. Using this approach several exact strong and adjacent dimensions are computed (for instance of the Petersen graph) and a positive answer is given to the Problem 4.1 of Fitzpatrick and Nowakowski [The strong isometric dimension of finite reflexive graphs, Discuss. Math. Graph Theory 20 (2000) 23–38] whether there is a graph G   with the strong isometric dimension bigger that ⌈|V(G)|/2⌉⌈|V(G)|/2⌉.