Linear Operators That Preserve the Genus of a Graph
A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.
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1969 ◽
Vol 16
(3)
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pp. 227-232
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1993 ◽
Vol 16
(4)
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pp. 823-824
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1969 ◽
Vol 21
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pp. 592-594
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1993 ◽
Vol 113
(1)
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pp. 161-172
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