35 research outputs found
Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs
For digraphs and , a homomorphism of to is a mapping $f:\
V(G)\dom V(H)uv\in A(G)f(u)f(v)\in A(H)u \in V(G)c_i(u), i \in V(H)f\sum_{u\in V(G)}c_{f(u)}(u)HHHGc_i(u)u\in V(G)i\in V(H)GH$ and, if one exists, to find one of minimum cost.
Minimum cost homomorphism problems encompass (or are related to) many well
studied optimization problems such as the minimum cost chromatic partition and
repair analysis problems. We focus on the minimum cost homomorphism problem for
locally semicomplete digraphs and quasi-transitive digraphs which are two
well-known generalizations of tournaments. Using graph-theoretic
characterization results for the two digraph classes, we obtain a full
dichotomy classification of the complexity of minimum cost homomorphism
problems for both classes
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 of Semicomplete Compositions
A strong arc decomposition of a digraph is a decomposition of its
arc set into two disjoint subsets and such that both of the
spanning subdigraphs 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 We
obtain a characterization of digraph compositions which
have a strong arc decomposition when is a semicomplete digraph and each
is an arbitrary digraph. Our characterization generalizes a
characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a
strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018)
on strong arc decompositions of digraph compositions in
which is semicomplete and each is arbitrary. Our proofs are
constructive and imply the existence of a polynomial algorithm for constructing
a \good{} decomposition of a digraph , with
semicomplete, whenever such a decomposition exists
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