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Planar wheel-like bricks
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An edge e in a matching covered graph G is removable if G-e is matching covered; a pair {e; f} of edges of G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, which was introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A brick is a nonbipartite matching covered graph without nontrivial tight cuts. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty conjectured that every planar wheel-like brick is an odd wheel. We present a proof of this conjecture in this paper.
Forward citations
Cited by 2 Pith papers
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Bricks that every removable edge is solitary
Every simple nonsolid brick in which every removable edge is solitary decomposes recursively by splicing odd wheels, and this decomposition cannot use K4 as the wheel factor.
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Wheel-like bricks and minimal matching covered graphs
Every wheel-like brick lies in a recursively defined family obtained by splicing odd wheels, and every minimal matching covered graph has minimum degree 2 or 3.
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