Maximum non-path-connected graphs (Q792336)

Let n and i be integers with \(n\geq 4\) and 2\(\leq i\leq n-1\). \textit{M. Lewin} [J. Comb. Theory, Ser. B 25, 245-257 (1978; Zbl 0328.05124)] has determined the smallest size of a connected n-graph without end-vertices which ensures the existence of a path of length i between every pair of distinct vertices of the graph. This paper determines all the connected n-graphs without end-vertices of maximum size which fail to have a path of length i between every pair of distinct vertices.