153 research outputs found
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
Graph Relations and Constrained Homomorphism Partial Orders
We consider constrained variants of graph homomorphisms such as embeddings,
monomorphisms, full homomorphisms, surjective homomorpshims, and locally
constrained homomorphisms. We also introduce a new variation on this theme
which derives from relations between graphs and is related to
multihomomorphisms. This gives a generalization of surjective homomorphisms and
naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs.
Both R-cores and R-cocores of graphs are unique up to isomorphism and can be
computed in polynomial time.
The theory of the graph homomorphism order is well developed, and from it we
consider analogous notions defined for orders induced by constrained
homomorphisms. We identify corresponding cores, prove or disprove universality,
characterize gaps and dualities. We give a new and significantly easier proof
of the universality of the homomorphism order by showing that even the class of
oriented cycles is universal. We provide a systematic approach to simplify the
proofs of several earlier results in this area. We explore in greater detail
locally injective homomorphisms on connected graphs, characterize gaps and show
universality. We also prove that for every the homomorphism order on
the class of line graphs of graphs with maximum degree is universal
On Linear Layouts of Graphs
In a total order of the vertices of a graph, two edges with no endpoint in common can be \emphcrossing, \emphnested, or \emphdisjoint. A \emphk-stack (respectively, \emphk-queue, \emphk-arch) \emphlayout of a graph consists of a total order of the vertices, and a partition of the edges into k sets of pairwise non-crossing (non-nested, non-disjoint) edges. Motivated by numerous applications, stack layouts (also called \emphbook embeddings) and queue layouts are widely studied in the literature, while this is the first paper to investigate arch layouts.\par Our main result is a characterisation of k-arch graphs as the \emphalmost (k+1)-colourable graphs; that is, the graphs G with a set S of at most k vertices, such that G S is (k+1)-colourable.\par In addition, we survey the following fundamental questions regarding each type of layout, and in the case of queue layouts, provide simple proofs of a number of existing results. How does one partition the edges given a fixed ordering of the vertices? What is the maximum number of edges in each type of layout? What is the maximum chromatic number of a graph admitting each type of layout? What is the computational complexity of recognising the graphs that admit each type of layout?\par A comprehensive bibliography of all known references on these topics is included. \pa
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
Stacks, Queues and Tracks: Layouts of Graph Subdivisions
A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. \par Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \pa
On the Plumbing Structure of Fibre Surfaces
This thesis is situated in the mathematical field of low-dimensional topology and is concerned with a particular class of knots and links known as fibred links. A link L in the three-sphere is fibred if its complement admits a fibration over the circle with fibres the interiors of Seifert surfaces for L. These surfaces, called fibre surfaces, are our main objects of study. For example, all torus links and all positive braid links are fibred. The simplest example of a fibred link is the Hopf link; its fibre surface is an annular band with a full twist. Using a geometric operation called plumbing, Hopf bands can be glued together so as to form more complicated fibre surfaces. The reverse operation, deplumbing, consists in removing a Hopf band from a fibre surface. It amounts to cutting that fibre surface along a proper arc.
By a theorem of Giroux and Goodman's, every fibre surface has a plumbing structure: it can be obtained from the standard disk by plumbing and deplumbing some number of Hopf bands. Our main results concern the (non-)uniqueness of the plumbing structure of a given fibre surface. Specifically, we study the cases of torus links and positive arborescent Hopf plumbings. Among these, we characterise the links arising from simple plane curve singularities by the finiteness of proper arcs that correspond to deplumbing a Hopf band
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