3,076 research outputs found
Tutte Short Exact Sequences of Graphs
We associate two modules, the -parking critical module and the toppling
critical module, to an undirected connected graph . We establish a
Tutte-like short exact sequence relating the modules associated to , an edge
contraction and edge deletion ( is a non-bridge). As
applications of these short exact sequences, we relate the vanishing of certain
combinatorial invariants (the number of acyclic orientations on connected
partition graphs satisfying a unique sink property) of to the equality of
corresponding invariants of and . We also obtain a short
proof of a theorem of Merino that the critical polynomial of a graph is an
evaluation of its Tutte polynomial.Comment: 40 pages, 3 figure
Enumerating Cyclic Orientations of a Graph
Acyclic and cyclic orientations of an undirected graph have been widely
studied for their importance: an orientation is acyclic if it assigns a
direction to each edge so as to obtain a directed acyclic graph (DAG) with the
same vertex set; it is cyclic otherwise. As far as we know, only the
enumeration of acyclic orientations has been addressed in the literature. In
this paper, we pose the problem of efficiently enumerating all the
\emph{cyclic} orientations of an undirected connected graph with vertices
and edges, observing that it cannot be solved using algorithmic techniques
previously employed for enumerating acyclic orientations.We show that the
problem is of independent interest from both combinatorial and algorithmic
points of view, and that each cyclic orientation can be listed with
delay time. Space usage is with an additional setup cost
of time before the enumeration begins, or with a setup cost of
time
Generating connected acyclic digraphs uniformly at random
We describe a simple algorithm based on a Markov chain process to generate
simply connected acyclic directed graphs over a fixed set of vertices. This
algorithm is an extension of a previous one, designed to generate acyclic
digraphs, non necessarily connected.Comment: 6 page
The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
We show that the 4-variable generating function of certain orientation
related parameters of an ordered oriented matroid is the evaluation at (x + u,
y+v) of its Tutte polynomial. This evaluation contains as special cases the
counting of regions in hyperplane arrangements and of acyclic orientations in
graphs. Several new 2-variable expansions of the Tutte polynomial of an
oriented matroid follow as corollaries.
This result hold more generally for oriented matroid perspectives, with
specific special cases the counting of bounded regions in hyperplane
arrangements or of bipolar acyclic orientations in graphs.
In corollary, we obtain expressions for the partial derivatives of the Tutte
polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table
Odd length: odd diagrams and descent classes
We define and study odd analogues of classical geometric and combinatorial objects
associated to permutations, namely odd Schubert varieties, odd diagrams, and odd
inversion sets. We show that there is a bijection between odd inversion sets of
permutations and acyclic orientations of the Turán graph, that the dimension of the
odd Schubert variety associated to a permutation is the odd length of the permutation,
and give several necessary conditions for a subset of [ n ] × [ n ] to be the odd diagram
of a permutation. We also study the sign-twisted generating function of the odd length
over descent classes of the symmetric groups
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