4 research outputs found
Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions
We show that every graph of maximum degree 3 can be represented as the
intersection graph of axis parallel boxes in three dimensions, that is, every
vertex can be mapped to an axis parallel box such that two boxes intersect if
and only if their corresponding vertices are adjacent. In fact, we construct a
representation in which any two intersecting boxes just touch at their
boundaries. Further, this construction can be realized in linear time
Local Boxicity, Local Dimension, and Maximum Degree
In this paper, we focus on two recently introduced parameters in the
literature, namely `local boxicity' (a parameter on graphs) and `local
dimension' (a parameter on partially ordered sets). We give an `almost linear'
upper bound for both the parameters in terms of the maximum degree of a graph
(for local dimension we consider the comparability graph of a poset). Further,
we give an time deterministic algorithm to compute a local box
representation of dimension at most for a claw-free graph, where
and denote the number of vertices and the maximum degree,
respectively, of the graph under consideration. We also prove two other upper
bounds for the local boxicity of a graph, one in terms of the number of
vertices and the other in terms of the number of edges. Finally, we show that
the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page
REPRESENTING A CUBIC GRAPH AS THE INTERSECTION GRAPH OF AXIS-PARALLEL BOXES IN THREE DIMENSIONS
We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries