Geodesic
Spanning caterpillars with bounded diameter
Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 2, pp. 111-118

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A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most cn^3/4 (at most n).
Keywords: distance, spaning tree 
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Faudree, Ralph; Gould, Ronald; Jacobson, Michael; Lesniak, Linda. Spanning caterpillars with bounded diameter. Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 2, pp. 111-118. http://geodesic.mathdoc.fr/item/DMGT_1995_15_2_a0/