Geodesic
Graphs Suppressible to an Edge
Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 201-204

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DOI

An application of graph theory to automatic traffic control [2] gave rise to the problem of deciding which connected graphs have points of degree 2 which can be successively suppressed until only a single edge remains. 
Harary, F.; Krarup, J.; Schwenk, A. Graphs Suppressible to an Edge. Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 201-204. doi: 10.4153/CMB-1972-036-5
@article{10_4153_CMB_1972_036_5,
     author = {Harary, F. and Krarup, J. and Schwenk, A.},
     title = {Graphs {Suppressible} to an {Edge}},
     journal = {Canadian mathematical bulletin},
     pages = {201--204},
     year = {1972},
     volume = {15},
     number = {2},
     doi = {10.4153/CMB-1972-036-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-036-5/}
}
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