Longer Cycles in Essentially 4-Connected Planar Graphs
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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).
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