516 research outputs found
On non-even digraphs and symplectic pairs
A digraph is called {\bf noneven} if it is possible to assign weights of
0,1 to its arcs so that contains no cycle of even weight. A noneven digraph
corresponds to one or more nonsingular sign patterns. Given an
sign pattern , a {\bf symplectic pair} in is a pair of matrices
such that , , and . An unweighted
digraph allows a matrix property if at least one of the sign patterns
whose digraph is allows . Thomassen characterized the noneven,
2-connected symmetric digraphs (i.e., digraphs for which the existence of arc
implies the existence of arc . In the first part of our paper,
we recall this characterization and use it to determine which strong symmetric
digraphs allow symplectic pairs. A digraph is called {\bf semi-complete}
if, for each pair of distinct vertices , at least one of the arcs
digraph. In the second part of our paper, we fill a gap in these two
characterizations and present and prove correct versions of the main theorems
involved. We then pr oceed to determine which digraphs from these classes allow
symplectic pairs. and exists in . Thomassen presented a
characterization of two classes of strong, noneven digraphs: the semi-complete
class and the class of digraphs for which each vertex has total degree which
exceeds or equals the size of the digraph. In the second part of our paper, we
fill a gap in these two characterizations and present and prove correct
versions of the main theorems involved. We then p oceed to determine which
digraphs from these classes allow symplectic pairs.Comment: 24 page
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
Worst-case efficient dominating sets in digraphs
Let . {\it Worst-case efficient dominating sets in digraphs} are
conceived so that their presence in certain strong digraphs
corresponds to that of efficient dominating sets in star graphs : The
fact that the star graphs form a so-called dense segmental neighborly
E-chain is reflected in a corresponding fact for the digraphs .
Related chains of graphs and open problems are presented as well.Comment: 13 pages, 3 figure
Spanning eulerian subdigraphs in semicomplete digraphs
A digraph is eulerian if it is connected and every vertex has its in-degree
equal to its out-degree.
Having a spanning eulerian subdigraph is thus a weakening of having a
hamiltonian cycle.
In this paper, we first characterize the pairs of a semicomplete
digraph and an arc such that has a spanning eulerian subdigraph
containing . In particular, we show that if is -arc-strong, then
every arc is contained in a spanning eulerian subdigraph.
We then characterize the pairs of a semicomplete digraph and an
arc such that has a spanning eulerian subdigraph avoiding . In
particular, we prove that every -arc-strong semicomplete digraph has a
spanning eulerian subdigraph avoiding any prescribed arc. We also prove the
existence of a (minimum) function such that every -arc-strong
semicomplete digraph contains a spanning eulerian subdigraph avoiding any
prescribed set of arcs: we prove , conjecture
and establish this conjecture for and when the arcs
that we delete form a forest of stars.
A digraph is eulerian-connected if for any two distinct vertices ,
the digraph has a spanning -trail. We prove that every
-arc-strong semicomplete digraph is eulerian-connected.
All our results may be seen as arc analogues of well-known results on
hamiltonian cycles in semicomplete digraphs
Characterizations and Directed Path-Width of Sequence Digraphs
Computing the directed path-width of a directed graph is an NP-hard problem.
Even for digraphs of maximum semi-degree 3 the problem remains hard. We propose
a decomposition of an input digraph G=(V,A) by a number k of sequences with
entries from V, such that (u,v) in A if and only if in one of the sequences
there is an occurrence of u appearing before an occurrence of v. We present
several graph theoretical properties of these digraphs. Among these we give
forbidden subdigraphs of digraphs which can be defined by k=1 sequence, which
is a subclass of semicomplete digraphs. Given the decomposition of digraph G,
we show an algorithm which computes the directed path-width of G in time
O(k\cdot (1+N)^k), where N denotes the maximum sequence length. This leads to
an XP-algorithm w.r.t. k for the directed path-width problem. Our result
improves the algorithms of Kitsunai et al. for digraphs of large directed
path-width which can be decomposed by a small number of sequence.Comment: 31 page
Packing and domination parameters in digraphs
Given a digraph , a set is a packing set in if
there are no arcs joining vertices of and for any two vertices
the sets of in-neighbors of and are disjoint. The set is a
dominating set (an open dominating set) in if every vertex not in (in
) has an in-neighbor in . Moreover, a dominating set is called a
total dominating set if the subgraph induced by has no isolated vertices.
The packing sets of maximum cardinality and the (total, open) dominating sets
of minimum cardinality in digraphs are studied in this article. We prove that
the two optimal sets concerning packing and domination achieve the same value
for directed trees, and give some applications of it. We also show analogous
equalities for all connected contrafunctional digraphs, and characterize all
such digraphs for which such equalities are satisfied. Moreover, sharp
bounds on the maximum and the minimum cardinalities of packing and dominating
sets, respectively, are given for digraphs. Finally, we present solutions for
two open problems, concerning total and open dominating sets of minimum
cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283--292]
Local Out-Tournaments with Upset Tournament Strong Components I: Full and Equal {0,1}-Matrix Ranks
A digraph D is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corresponding results for biclique covers and partitions of the digraph are provided
On Hamiltonian Bypasses in one Class of Hamiltonian Digraphs
Let be a strongly connected directed graph of order which
satisfies the following condition (*): for every pair of non-adjacent vertices
with a common in-neighbour and . In \cite{[2]} (J. of Graph Theory 22 (2) (1996) 181-187)) J.
Bang-Jensen, G. Gutin and H. Li proved that is Hamiltonian. In [9] it was
shown that if satisfies the condition (*) and the minimum semi-degree of
at least two, then either contains a pre-Hamiltonian cycle (i.e., a
cycle of length ) or is even and is isomorphic to the complete
bipartite digraph (or to the complete bipartite digraph minus one arc) with
partite sets of cardinalities of and . In this paper we show that if
the minimum out-degree of at least two and the minimum in-degree of at
least three, then contains also a Hamiltonian bypass, (i.e., a subdigraph
is obtained from a Hamiltonian cycle by reversing exactly one arc).Comment: 14 page
Out-colourings of Digraphs
We study vertex colourings of digraphs so that no out-neighbourhood is
monochromatic and call such a colouring an {\bf out-colouring}. The problem of
deciding whether a given digraph has an out-colouring with only two colours
(called a 2-out-colouring) is -complete. We show that for every choice of positive integers there
exists a -strong bipartite tournament which needs at least colours in
every out-colouring. Our main results are on tournaments and semicomplete
digraphs. We prove that, except for the Paley tournament , every strong
semicomplete digraph of minimum out-degree at least 3 has a 2-out-colouring.
Furthermore, we show that every semicomplete digraph on at least 7 vertices has
a 2-out-colouring if and only if it has a {\bf balanced} such colouring, that
is, the difference between the number of vertices that receive colour 1 and
colour 2 is at most one. In the second half of the paper we consider the
generalization of 2-out-colourings to vertex partitions of a
digraph so that each of the three digraphs induced by respectively, the
vertices of , the vertices of and all arcs between and
have minimum out-degree for a prescribed integer . Using
probabilistic arguments we prove that there exists an absolute positive
constant so that every semicomplete digraph of minimum out-degree at least
has such a partition. This is tight up to the value of
The Matrix of Maximum Out Forests of a Digraph and Its Applications
We study the maximum out forests of a (weighted) digraph and the matrix of
maximum out forests. A maximum out forest of a digraph G is a spanning subgraph
of G that consists of disjoint diverging trees and has the maximum possible
number of arcs. If a digraph contains any out arborescences, then maximum out
forests coincide with them. We provide a new proof to the Markov chain tree
theorem saying that the matrix of Ces`aro limiting probabilities of an
arbitrary stationary finite Markov chain coincides with the normalized matrix
of maximum out forests of the weighted digraph that corresponds to the Markov
chain. We discuss the applications of the matrix of maximum out forests and its
transposition, the matrix of limiting accessibilities of a digraph, to the
problems of preference aggregation, measuring the vertex proximity, and
uncovering the structure of a digraph.Comment: 27 pages, 3 figure
- …