TY - GEN
T1 - Contact Representations of Planar Graphs - Extending a Partial Representation is Hard
AU - Chaplick, Steven
AU - Dorbec, Paul
AU - Kratochvíl, Jan
AU - Montassier, Mickaël
AU - Stacho, Juraj
N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.
PY - 2014
Y1 - 2014
N2 - Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.On the positive side we give two polynomial time algorithms for the grid contact case. The first one is for the case when all vertical segments are pre-represented (note: the problem remains NP-complete when a subset of the vertical segments is specified, even if none of the horizontals are). Secondly, we show that the case when the vertical segments have only their x-coordinates specified (i.e., they are ordered horizontally) is polynomially equivalent to level planarity, which is known to be solvable in polynomial time.
AB - Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.On the positive side we give two polynomial time algorithms for the grid contact case. The first one is for the case when all vertical segments are pre-represented (note: the problem remains NP-complete when a subset of the vertical segments is specified, even if none of the horizontals are). Secondly, we show that the case when the vertical segments have only their x-coordinates specified (i.e., they are ordered horizontally) is polynomially equivalent to level planarity, which is known to be solvable in polynomial time.
U2 - 10.1007/978-3-319-12340-0_12
DO - 10.1007/978-3-319-12340-0_12
M3 - Conference article in proceeding
T3 - Lecture Notes in Computer Science
SP - 139
EP - 151
BT - Graph-Theoretic Concepts in Computer Science. WG 2014
A2 - Kratsch, D.
A2 - Todinca, I.
ER -