34 research outputs found
A study on Dicycles and Eulerian Subdigraphs
1. Dicycle cover of Hamiltonian oriented graphs. A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs, including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.;2. Supereulerian digraphs with given local structures . Catlin in 1988 indicated that there exist graph families F such that if every edge e in a graph G lies in a subgraph He of G isomorphic to a member in F, then G is supereulerian. In particular, if every edge of a connected graph G lies in a 3-cycle, then G is supereulerian. The purpose of this research is to investigate how Catlin\u27s theorem can be extended to digraphs. A strong digraph D is supereulerian if D contains a spanning eulerian subdigraph. We show that there exists an infinite family of non-supereulerian strong digraphs each arc of which lies in a directed 3-cycle. We also show that there exist digraph families H such that a strong digraph D is supereulerian if every arc a of D lies in a subdigraph Ha isomorphic to a member of H. A digraph D is symmetric if (x, y) ∈ A( D) implies (y, x) ∈ A( D); and is symmetrically connected if every pair of vertices of D are joined by a symmetric dipath. A digraph D is partially symmetric if the digraph obtained from D by contracting all symmetrically connected components is symmetrically connected. It is known that a partially symmetric digraph may not be symmetrically connected. We show that symmetrically connected digraphs and partially symmetric digraphs are such families. Sharpness of these results are discussed.;3. On a class of supereulerian digraphs. The 2-sum of two digraphs D1 and D2, denoted D1 ⊕2 D2, is the digraph obtained from the disjoint union of D 1 and D2 by identifying an arc in D1 with an arc in D2. A digraph D is supereulerian if D contains a spanning eulerian subdigraph. It has been noted that the 2-sum of two supereulerian (or even hamiltonian) digraphs may not be supereulerian. We obtain several sufficient conditions on D1 and D 2 for D1 ⊕2 D 2 to be supereulerian. In particular, we show that if D 1 and D2 are symmetrically connected or partially symmetric, then D1 ⊕2 D2 is supereulerian
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
Strong arc decompositions of split digraphs
A {\bf strong arc decomposition} of a digraph is a partition of its
arc set into two sets such that the digraph is
strong for . Bang-Jensen and Yeo (2004) conjectured that there is some
such that every -arc-strong digraph has a strong arc decomposition. They
also proved that with one exception on 4 vertices every 2-arc-strong
semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang
(2010) extended this result to locally semicomplete digraphs by proving that
every 2-arc-strong locally semicomplete digraph which is not the square of an
even cycle has a strong arc decomposition. This implies that every 3-arc-strong
locally semicomplete digraph has a strong arc decomposition. A {\bf split
digraph} is a digraph whose underlying undirected graph is a split graph,
meaning that its vertices can be partioned into a clique and an independent
set. Equivalently, a split digraph is any digraph which can be obtained from a
semicomplete digraph by adding a new set of vertices and some
arcs between and . In this paper we prove that every 3-arc-strong split
digraph has a strong arc decomposition which can be found in polynomial time
and we provide infinite classes of 2-strong split digraphs with no strong arc
decomposition. We also pose a number of open problems on split digraphs
Spanning eulerian subdigraphs in semicomplete digraphs
A digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a. In particular, we show that if D is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph avoiding a. In particular, we prove that every 2-arc-strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function f (k) such that every f (k)-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k arcs. We conjecture that f (k) = k + 1 and establish this conjecture for k ≤ 3 and when the k arcs that we delete form a forest of stars. A digraph D is eulerian-connected if for any two distinct vertices x, y, the digraph D has a spanning (x, y)-trail. We prove that every 2-arc-strong semicomplete digraph is eulerianconnected. All our results may be seen as arc analogues of well-known results on hamiltonian paths and cycles in semicomplete digraphs
Finding an induced subdivision of a digraph
We consider the following problem for oriented graphs and digraphs: Given an
oriented graph (digraph) , does it contain an induced subdivision of a
prescribed digraph ? The complexity of this problem depends on and on
whether must be an oriented graph or is allowed to contain 2-cycles. We
give a number of examples of polynomial instances as well as several
NP-completeness proofs
Spanning eulerian subdigraphs in semicomplete digraphs
International audienceA digraph is eulerian if it is connected and every vertex has its in-degree equal to its outdegree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph containing a. In particular, we show that if D is 2-arc-strong, then every arc is contained in a spanning eulerian subdigraph. We then characterize the pairs (D, a) of a semicomplete digraph D and an arc a such that D has a spanning eulerian subdigraph avoiding a. In particular, we prove that every 2-arc-strong semicomplete digraph has a spanning eulerian subdigraph avoiding any prescribed arc. We also prove the existence of a (minimum) function f (k) such that every f (k)-arc-strong semicomplete digraph contains a spanning eulerian subdigraph avoiding any prescribed set of k arcs. We conjecture that f (k) = k + 1 and establish this conjecture for k ≤ 3 and when the k arcs that we delete form a forest of stars. A digraph D is eulerian-connected if for any two distinct vertices x, y, the digraph D has a spanning (x, y)-trail. We prove that every 2-arc-strong semicomplete digraph is eulerianconnected. All our results may be seen as arc analogues of well-known results on hamiltonian paths and cycles in semicomplete digraphs