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let H be minor of a graph G. Is it always true that embedding surface of H is less than embedding surface of...
$begingroup$
let $H$ be minor of a graph $G$. Is it always true that (genus of embedding surface of $H$) is $leq$ (genus of embedding surface of $G$).
graph-theory planar-graph
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
$endgroup$
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$begingroup$
let $H$ be minor of a graph $G$. Is it always true that (genus of embedding surface of $H$) is $leq$ (genus of embedding surface of $G$).
graph-theory planar-graph
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
$endgroup$
add a comment |
$begingroup$
let $H$ be minor of a graph $G$. Is it always true that (genus of embedding surface of $H$) is $leq$ (genus of embedding surface of $G$).
graph-theory planar-graph
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
$endgroup$
let $H$ be minor of a graph $G$. Is it always true that (genus of embedding surface of $H$) is $leq$ (genus of embedding surface of $G$).
graph-theory planar-graph
graph-theory planar-graph
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
asked Jan 29 at 14:18
bandana pandeybandana pandey
63
63
add a comment |
add a comment |
1 Answer
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$begingroup$
Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.
answered Jan 29 at 14:27
quaraguequarague
547212
$endgroup$
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1 Answer
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1 Answer
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oldest
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active
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$begingroup$
Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.
answered Jan 29 at 14:27
quaraguequarague
547212
$endgroup$
add a comment |
$begingroup$
Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.
answered Jan 29 at 14:27
quaraguequarague
547212
$endgroup$
add a comment |
$begingroup$
Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.
answered Jan 29 at 14:27
quaraguequarague
547212
$endgroup$
Consider the embedding of $G$. I claim we can construct an embedding of $H$ on the same surface. For the edge deletions this is trivial. For the edge contractions one can picture the process as a continuous deformation of the graph without changing the surface. Hence any sequence of edge deletions and and edge contractions that builds $H$ from $G$ gives you an embedding of $H$ on the embedding surface of $G$. Therefore the embedding genus of $H$ has to be smaller or equal than the embedding genus of $G$.
answered Jan 29 at 14:27
quaraguequarague
547212
answered Jan 29 at 14:27
quaraguequarague
547212
answered Jan 29 at 14:27
quaraguequarague
547212
answered Jan 29 at 14:27
quaraguequarague
547212
547212
add a comment |
add a comment |
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