5 research outputs found
On the Rectangles Induced by Points
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\newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\|
{#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s} A set of
points in the plane, induces a set of Delaunay-type axis-parallel
rectangles , potentially of quadratic size, where an axis-parallel
rectangle is in , if it has two points of as corners, and no
other point of in it. We study various algorithmic problems related to this
set of rectangles, including how to compute it, in near linear time, and handle
various algorithmic tasks on it, such as computing its union and depth. The set
of rectangles induces the rectangle influence graph , which we also study. Potentially our most interesting result
is showing that this graph can be described as the union of bicliques,
where the total weight of the bicliques is . Here, the weight of
a bicliques is the cardinality of its vertices
Witness bar visibility graphs
Bar visibility graphs were introduced in the seventies as a model for some VLSI layout problems.
They have been also studied since then by the graph drawing community, and recently several
generalizations and restricted versions have been proposed.
We introduce a generalization, witness-bar visibility graphs, and we prove that this class encom-
passes all the bar-visibility variations considered so far. In addition, we show that many classes of graphs are contained in this family, including in particular all planar graphs, interval graphs, circular arc graphs and permutation graphsPeer ReviewedPostprint (published version
Mutual Witness Proximity Drawings of Isomorphic Trees
A pair of graphs admits a mutual witness proximity
drawing when: (i) represents
, and (ii) there is an edge in if and only if there is
no vertex in that is ``too close'' to both and
(). In this paper, we consider infinitely many definitions of closeness
by adopting the -proximity rule for any and study
pairs of isomorphic trees that admit a mutual witness -proximity
drawing. Specifically, we show that every two isomorphic trees admit a mutual
witness -proximity drawing for any . The
constructive technique can be made ``robust'': For some tree pairs we can
suitably prune linearly many leaves from one of the two trees and still retain
their mutual witness -proximity drawability. Notably, in the special
case of isomorphic caterpillars and , we construct linearly separable
mutual witness Gabriel drawings.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Witness rectangle graphs
In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.Peer ReviewedPostprint (published version
Witness rectangle graphs
In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.Peer Reviewe