94 research outputs found
Pixel and Voxel Representations of Graphs
We study contact representations for graphs, which we call pixel
representations in 2D and voxel representations in 3D. Our representations are
based on the unit square grid whose cells we call pixels in 2D and voxels in
3D. Two pixels are adjacent if they share an edge, two voxels if they share a
face. We call a connected set of pixels or voxels a blob. Given a graph, we
represent its vertices by disjoint blobs such that two blobs contain adjacent
pixels or voxels if and only if the corresponding vertices are adjacent. We are
interested in the size of a representation, which is the number of pixels or
voxels it consists of.
We first show that finding minimum-size representations is NP-complete. Then,
we bound representation sizes needed for certain graph classes. In 2D, we show
that, for -outerplanar graphs with vertices, pixels are
always sufficient and sometimes necessary. In particular, outerplanar graphs
can be represented with a linear number of pixels, whereas general planar
graphs sometimes need a quadratic number. In 3D, voxels are
always sufficient and sometimes necessary for any -vertex graph. We improve
this bound to for graphs of treewidth and to
for graphs of genus . In particular, planar graphs
admit representations with voxels
On Hypergraph Supports
Let be a hypergraph. A support is a graph
on such that for each , the subgraph of induced on the
elements in is connected. In this paper, we consider hypergraphs defined on
a host graph. Given a graph , with ,
and a collection of connected subgraphs of , a primal support
is a graph on such that for each , the
induced subgraph on vertices is connected. A \emph{dual support} is a graph on
s.t. for each , the induced subgraph
is connected, where . We present
sufficient conditions on the host graph and hyperedges so that the resulting
support comes from a restricted family.
We primarily study two classes of graphs: If the host graph has genus
and the hypergraphs satisfy a topological condition of being
\emph{cross-free}, then there is a primal and a dual support of genus at most
. If the host graph has treewidth and the hyperedges satisfy a
combinatorial condition of being \emph{non-piercing}, then there exist primal
and dual supports of treewidth . We show that this exponential blow-up
is sometimes necessary. As an intermediate case, we also study the case when
the host graph is outerplanar. Finally, we show applications of our results to
packing and covering, and coloring problems on geometric hypergraphs
A note on compact and compact circular edge-colorings of graphs
Graphs and Algorithm
Recommended from our members
The frequency assignment problem
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight.
We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR
- …