11,059 research outputs found
On powers of interval graphs and their orders
It was proved by Raychaudhuri in 1987 that if a graph power is an
interval graph, then so is the next power . This result was extended to
-trapezoid graphs by Flotow in 1995. We extend the statement for interval
graphs by showing that any interval representation of can be extended
to an interval representation of that induces the same left endpoint and
right endpoint orders. The same holds for unit interval graphs. We also show
that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the
main result of this note, follows from earlier results of [G. Agnarsson, P.
Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and
dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003].
This version is updated accordingl
Total Curvature of Graphs after Milnor and Euler
We define a new notion of total curvature, called net total curvature, for
finite graphs embedded in Rn, and investigate its properties. Two guiding
principles are given by Milnor's way of measuring the local crookedness of a
Jordan curve via a Crofton-type formula, and by considering the double cover of
a given graph as an Eulerian circuit. The strength of combining these ideas in
defining the curvature functional is (1) it allows us to interpret the
singular/non-eulidean behavior at the vertices of the graph as a superposition
of vertices of a 1-dimensional manifold, and thus (2) one can compute the total
curvature for a wide range of graphs by contrasting local and global properties
of the graph utilizing the integral geometric representation of the curvature.
A collection of results on upper/lower bounds of the total curvature on
isotopy/homeomorphism classes of embeddings is presented, which in turn
demonstrates the effectiveness of net total curvature as a new functional
measuring complexity of spatial graphs in differential-geometric terms.Comment: Most of the results contained in "Total curvature and isotopy of
graphs in ."(arXiv:0806.0406) have been incorporated into the current
articl
OBDD-Based Representation of Interval Graphs
A graph can be described by the characteristic function of the
edge set which maps a pair of binary encoded nodes to 1 iff the nodes
are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store
can lead to a (hopefully) compact representation. Given the OBDD as an
input, symbolic/implicit OBDD-based graph algorithms can solve optimization
problems by mainly using functional operations, e.g. quantification or binary
synthesis. While the OBDD representation size can not be small in general, it
can be provable small for special graph classes and then also lead to fast
algorithms. In this paper, we show that the OBDD size of unit interval graphs
is and the OBDD size of interval graphs is $O(\
| V \ | \log \ | V \ |)\Omega(\ | V \ | \log
\ | V \ |)O(\log \ | V \ |)O(\log^2 \ | V \ |)$ operations and
evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic
Concepts 201
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