20 research outputs found
On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admit an
intersection representation by L-shaped curves, where additionally the topmost
points of each curve are assumed to belong to a common horizontal line. We
prove that Grounded-L graphs admit an equivalent characterisation in terms of
vertex ordering with forbidden patterns.
We also compare this class to related intersection classes, such as the
grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance
graphs), or the outer-1-string graphs. We give constructions showing that these
classes are all distinct and satisfy only trivial or previously known
inclusions.Comment: 16 pages, 6 figure
On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns.
We also compare this class to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions
Coloring polygon visibility graphs and their generalizations
Curve pseudo-visibility graphs generalize polygon and pseudo-
polygon visibility graphs and form a hereditary class of
graphs. We prove that every curve pseudo-visibility graph
with clique number ω has chromatic number at most 3 · 4ω−1.
The proof is carried through in the setting of ordered graphs;
we identify two conditions satisfied by every curve pseudo-
visibility graph (considered as an ordered graph) and prove
that they are sufficient for the claimed bound. The proof is
algorithmic: both the clique number and a coloring with the
claimed number of colors can be computed in polynomial time
B-VPG Representation of AT-free Outerplanar Graphs
B-VPG graphs are intersection graphs of axis-parallel line segments in
the plane. In this paper, we show that all AT-free outerplanar graphs are
B-VPG. We first prove that every AT-free outerplanar graph is an induced
subgraph of a biconnected outerpath (biconnected outerplanar graphs whose weak
dual is a path) and then we design a B-VPG drawing procedure for
biconnected outerpaths. Our proofs are constructive and give a polynomial time
B-VPG drawing algorithm for the class.
We also characterize all subgraphs of biconnected outerpaths and name this
graph class "linear outerplanar". This class is a proper superclass of AT-free
outerplanar graphs and a proper subclass of outerplanar graphs with pathwidth
at most 2. It turns out that every graph in this class can be realized both as
an induced subgraph and as a spanning subgraph of (different) biconnected
outerpaths.Comment: A preliminary version, which did not contain the characterization of
linear outerplanar graphs (Section 3), was presented in the
International Conference on Algorithms and Discrete Applied Mathematics
(CALDAM) 2022. The definition of linear outerplanar graphs in this paper
differs from that in the preliminary version and hence Section 4 is ne
Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames
We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any epsilon>0, there exists a (2+epsilon)-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first O(1)-approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2+epsilon)-approximation for the problem with "diagonal-anchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017)
Colouring Polygon Visibility Graphs and Their Generalizations
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time
Recognizing Stick Graphs with and without Length Constraints
Stick graphs are intersection graphs of horizontal and vertical line segments
that all touch a line of slope -1 and lie above this line. De Luca et al.
[GD'18] considered the recognition problem of stick graphs when no order is
given (STICK), when the order of either one of the two sets is given (STICK_A),
and when the order of both sets is given (STICK_AB). They showed how to solve
STICK_AB efficiently.
In this paper, we improve the running time of their algorithm, and we solve
STICK_A efficiently. Further, we consider variants of these problems where the
lengths of the sticks are given as input. We show that these variants of STICK,
STICK_A, and STICK_AB are all NP-complete. On the positive side, we give an
efficient solution for STICK_AB with fixed stick lengths if there are no
isolated vertices