5,097 research outputs found
Recipe theorems for polynomial invariants on ribbon graphs with half-edges
We provide recipe theorems for the Bollob\`as and Riordan polynomial
defined on classes of ribbon graphs with half-edges introduced in
arXiv:1310.3708[math.GT]. We also define a generalized transition polynomial
on this new category of ribbon graphs and establish a relationship between
and .Comment: 24 pages, 14 figure
Chord Diagrams and Gauss Codes for Graphs
Chord diagrams on circles and their intersection graphs (also known as circle
graphs) have been intensively studied, and have many applications to the study
of knots and knot invariants, among others. However, chord diagrams on more
general graphs have not been studied, and are potentially equally valuable in
the study of spatial graphs. We will define chord diagrams for planar
embeddings of planar graphs and their intersection graphs, and prove some basic
results. Then, as an application, we will introduce Gauss codes for immersions
of graphs in the plane and give algorithms to determine whether a particular
crossing sequence is realizable as the Gauss code of an immersed graph.Comment: 20 pages, many figures. This version has been substantially
rewritten, and the results are stronge
Tree-like properties of cycle factorizations
We provide a bijection between the set of factorizations, that is, ordered
(n-1)-tuples of transpositions in whose product is (12...n),
and labelled trees on vertices. We prove a refinement of a theorem of
D\'{e}nes that establishes new tree-like properties of factorizations. In
particular, we show that a certain class of transpositions of a factorization
correspond naturally under our bijection to leaf edges of a tree. Moreover, we
give a generalization of this fact.Comment: 10 pages, 3 figure
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
The slopes determined by n points in the plane
Let , , ..., be the slopes of the
lines connecting points in general position in the plane. The ideal
of all algebraic relations among the defines a configuration space
called the {\em slope variety of the complete graph}. We prove that is
reduced and Cohen-Macaulay, give an explicit Gr\"obner basis for it, and
compute its Hilbert series combinatorially. We proceed chiefly by studying the
associated Stanley-Reisner simplicial complex, which has an intricate recursive
structure. In addition, we are able to answer many questions about the geometry
of the slope variety by translating them into purely combinatorial problems
concerning enumeration of trees.Comment: 36 pages; final published versio
On the Kontsevich integral for knotted trivalent graphs
We construct an extension of the Kontsevich integral of knots to knotted
trivalent graphs, which commutes with orientation switches, edge deletions,
edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first
constructed such an extension, building on Drinfel'd's theory of associators.
We construct a step by step definition, using elementary Kontsevich integral
methods, to get a one-parameter family of corrections that all yield invariants
well behaved under the graph operations above.Comment: Journal version, 47 page
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