7 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) β A( D) implies (y, x) β 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 β2 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 β2 D 2 to be supereulerian. In particular, we show that if D 1 and D2 are symmetrically connected or partially symmetric, then D1 β2 D2 is supereulerian
Vertex-Pancyclism in the Generalized Sum of Digraphs
A digraph , of order is pancyclic, whenever
contains a directed cycle of length for each ; and
is vertex-pancyclic iff, for each vertex and each , contains a directed cycle of length passing through
. Let be a collection of pairwise vertex disjoint
digraphs. The generalized sum (g.s.) of , denoted by
or , is the set of
all digraphs satisfying: (i) , (ii) for , and (iii) for each pair of
vertices belonging to different summands of , there is exactly one arc
between them, with an arbitrary but fixed direction. A digraph in
will be called a generalized sum (g.s.) of . Let be a collection of pairwise
vertex disjoint Hamiltonian digraphs, in this paper we give simple sufficient
conditions for a digraph be vertex-pancyclic. This
result extends a result obtained by Cordero-Michel, Galeana-S\'anchez and
Goldfeder in 2016.Comment: 13 pages, 5 figure
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves