9 research outputs found
Planar graphs as L-intersection or L-contact graphs
The L-intersection graphs are the graphs that have a representation as
intersection graphs of axis parallel shapes in the plane. A subfamily of these
graphs are {L, |, --}-contact graphs which are the contact graphs of axis
parallel L, |, and -- shapes in the plane. We prove here two results that were
conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are
L-intersection graphs, and that triangle-free planar graphs are {L, |,
--}-contact graphs. These results are obtained by a new and simple
decomposition technique for 4-connected triangulations. Our results also
provide a much simpler proof of the known fact that planar graphs are segment
intersection graphs
EPG-representations with small grid-size
In an EPG-representation of a graph each vertex is represented by a path
in the rectangular grid, and is an edge in if and only if the paths
representing an share a grid-edge. Requiring paths representing edges
to be x-monotone or, even stronger, both x- and y-monotone gives rise to three
natural variants of EPG-representations, one where edges have no monotonicity
requirements and two with the aforementioned monotonicity requirements. The
focus of this paper is understanding how small a grid can be achieved for such
EPG-representations with respect to various graph parameters.
We show that there are -edge graphs that require a grid of area
in any variant of EPG-representations. Similarly there are
pathwidth- graphs that require height and area in
any variant of EPG-representations. We prove a matching upper bound of
area for all pathwidth- graphs in the strongest model, the one where edges
are required to be both x- and y-monotone. Thus in this strongest model, the
result implies, for example, , and area bounds
for bounded pathwidth graphs, bounded treewidth graphs and all classes of
graphs that exclude a fixed minor, respectively. For the model with no
restrictions on the monotonicity of the edges, stronger results can be achieved
for some graph classes, for example an area bound for bounded treewidth
graphs and bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Polynomial-Time Approximation Schemes for Independent Packing Problems on Fractionally Tree-Independence-Number-Fragile Graphs
We investigate a relaxation of the notion of treewidth-fragility, namely tree-independence-number-fragility. In particular, we obtain polynomial-time approximation schemes for independent packing problems on fractionally tree-independence-number-fragile graph classes. Our approach unifies and extends several known polynomial-time approximation schemes on seemingly unrelated graph classes, such as classes of intersection graphs of fat objects in a fixed dimension or proper minor-closed classes. We also study the related notion of layered tree-independence number, a relaxation of layered treewidth
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum