43,971 research outputs found
Three Existence Problems in Extremal Graph Theory
Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics.
In this thesis we take this approach to three different structural questions rooted in extremal graph theory.
When studying graph representations, we seek efficient ways to encode the structure of a graph.
For example, an {\it interval representation} of a graph is an assignment of intervals on the real line to the vertices of such that two vertices are adjacent if and only if their intervals intersect.
We consider graphs that have {\it bar -visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations.
We obtain results on , the family of graphs having bar -visibility representations.
We also study .
In particular, we determine the largest complete graph having a bar -visibility representation, and we show that there are graphs that do not have bar -visibility representations for any .
Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs.
Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around provided that each move transfers weight from a vertex to a heavier neighbor.
Our goal in making acquisition moves is to consolidate all of the weight in on the minimum number of vertices; this minimum number is the {\it acquisition number} of .
We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight.
We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter .
We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex.
Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects.
Some local conditions are so limiting that very few objects satisfy the requirements.
For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor.
Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex.
We study a related structural restriction where similar phenomena occur.
For a fixed graph , we consider those graphs that do not contain and such that the addition of any edge completes exactly one copy of .
Such a graph is called {\it uniquely -saturated}.
We study the existence of uniquely -saturated graphs when is a path or a cycle.
In particular, we determine all of the uniquely -saturated graphs; there are exactly ten.
Interestingly, the uniquely -saturated graphs are precisely the friendship graphs characterized by Wilf
Bar 1-Visibility Drawings of 1-Planar Graphs
A bar 1-visibility drawing of a graph is a drawing of where each
vertex is drawn as a horizontal line segment called a bar, each edge is drawn
as a vertical line segment where the vertical line segment representing an edge
must connect the horizontal line segments representing the end vertices and a
vertical line segment corresponding to an edge intersects at most one bar which
is not an end point of the edge. A graph is bar 1-visible if has a bar
1-visibility drawing. A graph is 1-planar if has a drawing in a
2-dimensional plane such that an edge crosses at most one other edge. In this
paper we give linear-time algorithms to find bar 1-visibility drawings of
diagonal grid graphs and maximal outer 1-planar graphs. We also show that
recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs
are bar 1-visible graphs.Comment: 15 pages, 9 figure
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs
A graph is called a strong (resp. weak) bar 1-visibility graph if its
vertices can be represented as horizontal segments (bars) in the plane so that
its edges are all (resp. a subset of) the pairs of vertices whose bars have a
-thick vertical line connecting them that intersects at most one
other bar.
We explore the relation among weak (resp. strong) bar 1-visibility graphs and
other nearly planar graph classes. In particular, we study their relation to
1-planar graphs, which have a drawing with at most one crossing per edge;
quasi-planar graphs, which have a drawing with no three mutually crossing
edges; the squares of planar 1-flow networks, which are upward digraphs with
in- or out-degree at most one. Our main results are that 1-planar graphs and
the (undirected) squares of planar 1-flow networks are weak bar 1-visibility
graphs and that these are quasi-planar graphs
- …