Geodesic
Connectivity of path graphs
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 181-195.

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We prove a necessary and sufficient condition under which a connected graph has a connected P₃-path graph. Moreover, an analogous condition for connectivity of the Pₖ-path graph of a connected graph which does not contain a cycle of length smaller than k+1 is derived.
Keywords: connectivity, path graph, cycle 
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Knor, Martin; Niepel, L'udovít. Connectivity of path graphs. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 181-195. http://geodesic.mathdoc.fr/item/DMGT_2000_20_2_a2/

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