99,070 research outputs found
Cycles in Oriented 3-graphs
An oriented 3-graph consists of a family of triples (3-sets), each of which
is given one of its two possible cyclic orientations. A cycle in an oriented
3-graph is a positive sum of some of the triples that gives weight zero to each
2-set.
Our aim in this paper is to consider the following question: how large can
the girth of an oriented 3-graph (on vertices) be? We show that there exist
oriented 3-graphs whose shortest cycle has length : this
is asymptotically best possible. We also show that there exist 3-tournaments
whose shortest cycle has length , in complete contrast
to the case of 2-tournaments.Comment: 12 page
Enumeration of Preferred Extensions in Almost Oriented Digraphs
In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on n vertices, all preferred extensions can be enumerated in O^*(3^{n/3}) time and there are directed graphs with Omega(3^{n/3}) preferred extensions. We give faster enumeration algorithms for directed graphs with at most 0.8004 * n vertices occurring in 2-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time O(1.2321^n), and we show that there are oriented graphs with Omega(3^{n/6}) > Omega(1.2009^n) preferred extensions.
A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in 2-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017)
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
Embedding oriented graphs in books
A book consists of a line L in [special characters omitted]3, called the spine, and a collection of half planes, called pages, whose common boundary is L. A k-book is book with k pages. A k-page book embedding is a continuous one-to-one mapping of a graph G into a book such that the vertices are mapped into L and the edges are each mapped to either the spine or a particular page, such that no two edges cross in any page. Each page contains a planar subgraph of G. The book thickness, denoted bt(G), is the minimum number of pages for a graph to have a k-page book embedding. We focus on oriented graphs, and propose a new way to embed oriented graphs into books, called an oriented book embedding, and define oriented book thickness .
We investigate oriented graphs having oriented book thickness k using k-page critical oriented graphs, oriented graphs with oriented book thickness equal to k, but, for each arc, the deletion of that arc yields an oriented graph with oriented book thickness equal to k –1. We discuss several classes of two-page critical oriented graphs, and use them to characterize oriented graphs with oriented book thickness equal to one that are strictly uni-dicyclic graphs, oriented graphs having exactly one cycle, which is a directed cycle. We give a similar result for strictly bi-dicyclic graphs, oriented graphs having exactly two cycles, which are directed cycles
The smallest 5-chromatic tournament
A coloring of a digraph is a partition of its vertex set such that each class
induces a digraph with no directed cycles. A digraph is -chromatic if is
the minimum number of classes in such partition, and a digraph is oriented if
there is at most one arc between each pair of vertices. Clearly, the smallest
-chromatic digraph is the complete digraph on vertices, but determining
the order of the smallest -chromatic oriented graphs is a challenging
problem. It is known that the smallest -, - and -chromatic oriented
graphs have , and vertices, respectively. In 1994, Neumann-Lara
conjectured that a smallest -chromatic oriented graph has vertices. We
solve this conjecture and show that the correct order is
Packing Arc-Disjoint 4-Cycles in Oriented Graphs
Given a directed graph G and a positive integer k, the Arc Disjoint r-Cycle Packing problem asks whether G has k arc-disjoint r-cycles. We show that, for each integer r ? 3, Arc Disjoint r-Cycle Packing is NP-complete on oriented graphs with girth r. When r is even, the same result holds even when the input class is further restricted to be bipartite. On the positive side, focusing on r = 4 in oriented graphs, we study the complexity of the problem with respect to two parameterizations: solution size and vertex cover size. For the former, we give a cubic kernel with quadratic number of vertices. This is smaller than the compression size guaranteed by a reduction to the well-known 4-Set Packing. For the latter, we show fixed-parameter tractability using an unapparent integer linear programming formulation of an equivalent problem
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