Embeddings of Product Graphs Where One Factor is a Hypercube

Voltage graph theory can be used to describe embeddings of product graphs if one factor is a Cayley graph. We use voltage graphs to explore embeddings of various products where one factor is a hypercube, describing some minimal and symmetrical embeddings. We then define a graph product, the weak symmetric difference, and illustrate a voltage graph construction useful for obtaining an embedding of the weak symmetric difference of an arbitrary graph with a hypercube

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Enumeration of polyhedral graphs

This thesis is concerned with the design of a polyhedron enumeration algorithm. The approach taken focuses on specic classes of polyhedra and their graph theoretic properties. This is then compared more broadly to other graph enumeration algorithms that are concerned with the same or a superset which includes these properties. An original and novel algorithm is contributed to this area. The approach taken divides the problem into prescribed vertex and face degree sequences for the graphs. Using a range of existence, ordered enumeration and isomorphism techniques, it finds all unique 4-regular, 3-connected planar graphs. The algorithm is a vertex addition algorithm which means that each result output at a given stage has a new vertex added. Other results from different stages are never required for further computation and comparison, hence the process is embarrassingly parallel. Therefore, the enumeration can be distributed optimally across a cluster of computers. This work has led to a successfully implemented algorithm which takes a different approach to its treatment of the class of 4-regular, 3-connected planar graphs. As such this has led to observations and theory about other classes of graphs and graph embeddings which relate to this research