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Crossed Graph


A crossed graph is the name given by Brandenburg (2021) to a graph obtained from a polyhedron by drawing edges between every pair of vertices in all its faces. This process leaves triangular faces unchanged.

The following table summarizes some special cases.

base polyhedroncrossed graphreference
cube16-cell graph
cuboctahedronoctahedral line graph
elongated square gyrorocupolaelongated square gyrorocupola graphFabrici and Madaras (2007)
regular dodecahedroncrossed dodecahedral graphBrandenburg (2021)
icosidodecahedronicosahedral line graph
rhombic dodecahedroncrossed rhombic dodecahedron
truncated tetrahedroncirculant graph Ci_(12)(1,2,4,5,6)
CrossedDodecahedralGraph

The crossed dodecahedral graph is the graph obtained by crossing the regular dodecahedron. It can therefore be constructed by adding edges in a pentagrammatic configuration to each face of a regular dodecahedron. It is also the graph square of the dodecahedral graph. This graph is 2-planar, does not admit a straight-line 2-planar drawing, and has unique 2-planar embedding illustrated above (Brandenburg 2021, Bekos et al. 2017). It is implemented in the Wolfram Language as GraphData["CrossedDodecahedralGraph"].

A different use of "crossed graph" appears in the notion of the crossed prism graph adopted in this work.


See also

Crossed Prism Graph, Polyhedron, Polyhedron Face, Skeleton

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References

Bekos, M. A.; Kaufmann, M.; and Raftopoulou, C. N. "On Optimal 2- and 3-Planar Graphs." In SoCG 2017 (Ed. B. Aronov and M. J. Katz). Vol. 77 of LIPIcs, Schloss Dagstuhl--Leibniz-Zentrum für Informatik, pp. 16:1-16:16, 2017.Brandenburg, F. J. "Straight-Line Drawings of 1-Planar Graphs." 3 Sep 2021. https://arxiv.org/abs/2109.01692.Fabrici, I. and Madaras, T. "The Structure of 1-Planar Graphs." Disc. Math. 307, 854-865, 2007.Ringel

Cite this as:

Weisstein, Eric W. "Crossed Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CrossedGraph.html