Geodesic
Longer Cycles in Essentially 4-Connected Planar Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 269-277

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A planar 3-connected graph G is called essentially 4-connected if, for every 3-separator S, at least one of the two components of G − S is an isolated vertex. Jackson and Wormald proved that the length circ (G) of a longest cycle of any essentially 4-connected planar graph G on n vertices is at least 2n+4 /5 and Fabrici, Harant and Jendrol’ improved this result to circ (G) ≥ 1/2 (n+4). In the present paper, we prove that an essentially 4-connected planar graph on n vertices contains a cycle of length at least 3/5 (n+2) and that such a cycle can be found in time O(n^2).
Keywords: essentially 4-connected planar graph, longest cycle, circumference, shortness coefficient 
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Fabrici, Igor; Harant, Jochen; Mohr, Samuel; Schmidt, Jens M. Longer Cycles in Essentially 4-Connected Planar Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 269-277. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a17/