419 research outputs found
Isomorphism of graph classes related to the circular-ones property
We give a linear-time algorithm that checks for isomorphism between two 0-1
matrices that obey the circular-ones property. This algorithm leads to
linear-time isomorphism algorithms for related graph classes, including Helly
circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and
convex-round graphs.Comment: 25 pages, 9 figure
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
FO Model Checking of Geometric Graphs
Over the past two decades the main focus of research into first-order (FO)
model checking algorithms has been on sparse relational structures -
culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model
checking of nowhere dense classes of graphs. On contrary to that, except the
case of locally bounded clique-width only little is currently known about FO
model checking of dense classes of graphs or other structures. We study the FO
model checking problem for dense graph classes definable by geometric means
(intersection and visibility graphs). We obtain new nontrivial FPT results,
e.g., for restricted subclasses of circular-arc, circle, box, disk, and
polygon-visibility graphs. These results use the FPT algorithm by Gajarsk\'y et
al. for FO model checking of posets of bounded width. We also complement the
tractability results by related hardness reductions
Optimal Circular Arc Representations: Properties, Recognition, and Construction
AbstractWe investigate some properties of minimal interval and circular arc representations and give several optimal sequential and parallel recognition and construction algorithms. We show that, among other things, given ans×tinterval or circular arc representation matrix, •deciding if the representation is minimal can be done inO(logs) time withO(st/logs) EREW PRAM processors, or inO(1) time withO(st) common CRCW PRAM processors; •constructing an equivalent minimum interval representation can be done inO(log(st)) time withO(st/log(st)) EREW PRAM processors, or inO(logt/loglogt) time withO(stloglogt/logt) common CRCW PRAM processors, or inO(1) time withO(st) BSR processors; •constructing an equivalent minimal circular arc representation can be done inO(st) time
- …