@article{Slodkowski_Huber_2021, place={Houston, U.S.}, title={Contractible Edges and Peripheral Cycles in 3-Connected Graphs}, volume={10}, url={https://www.jsr.org/index.php/path/article/view/1193}, DOI={10.47611/jsr.v10i2.1193}, abstractNote={<p style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><span style="font-size: small;">Peripheral cycles (induced non-separating cycles) in a general 3-connected graph are analogous to the faces of a polyhedron. Using the works of various authors, this paper explores the distribution of contractible edges in 3-connected graphs as needed to prove a major result originally by Tutte: each edge in a 3-connected graph is part of at least 2 peripheral cycles that share only the edge and its end vertices. A complete, alternative proof of this theorem is provided. The inductive step is generalized into a new independent lemma, which states that each edge in a 3-connected graph with a non-adjacent contractible edge has at least as many peripheral cycles as in the contracted one.</span></span></p>}, number={2}, journal={Journal of Student Research}, author={Slodkowski, Alexander and Huber, Timothy}, year={2021}, month={Jul.} }