1,406 research outputs found
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth
We investigate crossing minimization for 1-page and 2-page book drawings. We
show that computing the 1-page crossing number is fixed-parameter tractable
with respect to the number of crossings, that testing 2-page planarity is
fixed-parameter tractable with respect to treewidth, and that computing the
2-page crossing number is fixed-parameter tractable with respect to the sum of
the number of crossings and the treewidth of the input graph. We prove these
results via Courcelle's theorem on the fixed-parameter tractability of
properties expressible in monadic second order logic for graphs of bounded
treewidth.Comment: Graph Drawing 201
Fractional colorings of partial -trees with no large clique
Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017]
asked, what is the largest chromatic number attainable by a graph of treewidth
with no subgraph? In this paper, we consider the fractional version
of this question. We prove that if has treewidth and clique number , then , and we
show that this bound is tight for . We also show that for each
value , there exists a graph of a large treewidth
and clique number satisfying .Comment: 9 page
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
A coarse geometric approach to graph layout problems
We define a range of new coarse geometric invariants based on various
graph-theoretic measures of complexity for finite graphs, including: treewidth,
pathwidth, cutwidth, search number, topological bandwidth, bandwidth, minimal
linear arrangment, sumcut, profile, vertex and edge separation. We prove that,
for bounded degree graphs, these invariants can be used to define functions
which satisfy a strong monotonicity property, namely they are monotonically
non-decreasing with respect to regular maps, and as such have potential
applications in coarse geometry and geometric group theory. On the
graph-theoretic side, we prove asymptotically optimal upper bounds on the
treewidth, pathwidth, cutwidth, search number, topological bandwidth, vertex
separation, edge separation, minimal linear arrangement, sumcut and profile for
the family of all finite subgraphs of any bounded degree graph whose separation
profile is known to be of the form for some . This large
class includes the Diestel-Leader graph, all Cayley graphs of non-virtually
cyclic polycyclic groups, uniform lattices in almost all connected unimodular
Lie groups, and certain hyperbolic groups.Comment: 19 page
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