21 research outputs found
Polynomial expansion and sublinear separators
Let be a class of graphs that is closed under taking subgraphs.
We prove that if for some fixed , every -vertex graph of
has a balanced separator of order , then any
depth- minor (i.e. minor obtained by contracting disjoint subgraphs of
radius at most ) of a graph in has average degree . This confirms a conjecture of Dvo\v{r}\'ak
and Norin.Comment: 6 pages, no figur
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
Modularity of minor-free graphs
We prove that a class of graphs with an excluded minor and with the maximum
degree sublinear in the number of edges is maximally modular, that is,
modularity tends to 1 as the number of edges tends to infinity.Comment: 7 pages, 1 figur
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Modularity of regular and treelike graphs
Clustering algorithms for large networks typically use modularity values to
test which partitions of the vertex set better represent structure in the data.
The modularity of a graph is the maximum modularity of a partition. We consider
the modularity of two kinds of graphs.
For -regular graphs with a given number of vertices, we investigate the
minimum possible modularity, the typical modularity, and the maximum possible
modularity. In particular, we see that for random cubic graphs the modularity
is usually in the interval , and for random -regular graphs
with large it usually is of order . These results help to
establish baselines for statistical tests on regular graphs.
The modularity of cycles and low degree trees is known to be close to 1: we
extend these results to `treelike' graphs, where the product of treewidth and
maximum degree is much less than the number of edges. This yields for example
the (deterministic) lower bound mentioned above on the modularity of
random cubic graphs.Comment: 25 page
Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs
Let C and D be hereditary graph classes. Consider the following problem: given a graph G in D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:
- the graphs in C are sparse, i.e., they have linearly many edges in terms of the number of vertices;
- the graphs in D admit balanced separators of size governed by their density, e.g., O(Delta) or O(sqrt{m}), where Delta and m denote the maximum degree and the number of edges, respectively; and
- the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph.
This leads, for example, to the following corollaries for specific classes C and D:
- a largest induced forest in a P_t-free graph can be found in 2^{O~(n^{2/3})} time, for every fixed t; and
- a largest induced planar graph in a string graph can be found in 2^{O~(n^{3/4})} time