42 research outputs found
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
A tourist guide through treewidth
A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography
Approximating pathwidth for graphs of small treewidth
We describe a polynomial-time algorithm which, given a graph with
treewidth , approximates the pathwidth of to within a ratio of
. This is the first algorithm to achieve an
-approximation for some function .
Our approach builds on the following key insight: every graph with large
pathwidth has large treewidth or contains a subdivision of a large complete
binary tree. Specifically, we show that every graph with pathwidth at least
has treewidth at least or contains a subdivision of a complete
binary tree of height . The bound is best possible up to a
multiplicative constant. This result was motivated by, and implies (with
), the following conjecture of Kawarabayashi and Rossman (SODA'18): there
exists a universal constant such that every graph with pathwidth
has treewidth at least or contains a subdivision of a
complete binary tree of height .
Our main technical algorithm takes a graph and some (not necessarily
optimal) tree decomposition of of width in the input, and it computes
in polynomial time an integer , a certificate that has pathwidth at
least , and a path decomposition of of width at most . The
certificate is closely related to (and implies) the existence of a subdivision
of a complete binary tree of height . The approximation algorithm for
pathwidth is then obtained by combining this algorithm with the approximation
algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth