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Published in:
2011 | OriginalPaper | Chapter
Point-Set Embeddings of Plane 3-Trees
(Extended Abstract)
Authors : Rahnuma Islam Nishat, Debajyoti Mondal, Md. Saidur Rahman
Published in: Graph Drawing
Publisher: Springer Berlin Heidelberg
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A straight-line drawing of a plane graph
G
is a planar drawing of
G
, where each vertex is drawn as a point and each edge is drawn as a straight-line segment. Given a set
S
of
n
points on the Euclidean plane, a point-set embedding of a plane graph
G
with
n
vertices on
S
is a straight-line drawing of
G
, where each vertex of
G
is mapped to a distinct point of
S
. The problem of deciding if
G
admits a point-set embedding on
S
is NP-complete in general and even when
G
is 2-connected and 2-outerplanar. In this paper we give an
O
(
n
2
log
n
) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω(
n
log
n
) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. Moreover, we consider a variant of the problem where we are given a plane 3-tree
G
with
n
vertices and a set
S
of
k
>
n
points, and give a polynomial time algorithm to find a point-set embedding of
G
on
S
if it exists.