322 research outputs found
Packing Strong Subgraph in Digraphs
In this paper, we study two types of strong subgraph packing problems in
digraphs, including internally disjoint strong subgraph packing problem and
arc-disjoint strong subgraph packing problem. These problems can be viewed as
generalizations of the famous Steiner tree packing problem and are closely
related to the strong arc decomposition problem. We first prove the
NP-completeness for the internally disjoint strong subgraph packing problem
restricted to symmetric digraphs and Eulerian digraphs. Then we get
inapproximability results for the arc-disjoint strong subgraph packing problem
and the internally disjoint strong subgraph packing problem. Finally we study
the arc-disjoint strong subgraph packing problem restricted to digraph
compositions and obtain some algorithmic results by utilizing the structural
properties
Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs
A digraph has a good pair at a vertex if has a pair of
arc-disjoint in- and out-branchings rooted at . Let be a digraph with
vertices and let be digraphs such that
has vertices Then the composition
is a digraph with vertex set and arc set
When is arbitrary, we obtain the following result: every strong digraph
composition in which for every , has a good pair
at every vertex of The condition of in this result cannot be
relaxed. When is semicomplete, we characterize semicomplete compositions
with a good pair, which generalizes the corresponding characterization by
Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As
a result, we can decide in polynomial time whether a given semicomplete
composition has a good pair rooted at a given vertex
Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs
A digraph has a good decomposition if has two disjoint sets
and such that both and are strong. Let be a
digraph with vertices and let be digraphs
such that has vertices Then the
composition is a digraph with vertex set
and arc set
For digraph compositions , we obtain sufficient
conditions for to have a good decomposition and a characterization of
with a good decomposition when is a strong semicomplete digraph and each
is an arbitrary digraph with at least two vertices.
For digraph products, we prove the following: (a) if is an integer
and is a strong digraph which has a collection of arc-disjoint cycles
covering all vertices, then the Cartesian product digraph (the
th powers with respect to Cartesian product) has a good decomposition; (b)
for any strong digraphs , the strong product has a good
decomposition
Revealed Cores: Characterizations and Structure
Characterizations of the choice functions that select the cores or the externally stable cores induced by an underlying revealed dominance digraph are provided. Relying on such characterizations, the basic order-theoretic structure of the corresponding sets of revealed cores is also analyzedCore, choice functions, dominance digraphs, revealed preference
A new family of posets generalizing the weak order on some Coxeter groups
We construct a poset from a simple acyclic digraph together with a valuation
on its vertices, and we compute the values of its M\"obius function. We show
that the weak order on Coxeter groups of type A, B, affine A, and the flag weak
order on the wreath product introduced by Adin,
Brenti and Roichman, are special instances of our construction. We conclude by
associating a quasi-symmetric function to each element of these posets. In the
and cases, this function coincides respectively with the
classical Stanley symmetric function, and with Lam's affine generalization
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