A flippable edge in an acyclic digraph is an edge whose reorientation leaves the graph acyclic. We characterize the spanning trees T of an undirected graph G such that there exists an acyclic orientation of G whose set of flippable edges is T . In particular for every edge e ∈ E(G) we give a linear algorithm returning an acyclic orientation and a spanning tree T containing e such that T is the ...

Given an embedded planar acyclic digraph G, we define the problem of"acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM)"to be the problem of determining an hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian digraph. Our results include: --We pro...

Given an embedded planar acyclic digraph G, the acyclic hamiltonian path completion with crossing minimization (AcyclicHPCCM) problem is to determine a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. In this paper, we present a linear time algorithm whi...

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that...

An acyclic homomorphism of a digraph D into a digraph F is a mapping φ : V (D) → V (F ) such that for every arc uv ∈ E(D), either φ(u) = φ(v) or φ(u)φ(v) is an arc of F , and for every vertex v ∈ V (F ), the subgraph of D induced on φ(v) is acyclic. For each fixed digraph F we consider the following decision problem: Does a given input digraph D admit an acyclic homomorphism to F? We prove that...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math. 11 (1959), 34–38], it was shown prob...

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...

Partial clique covering is one of the most basic coding schemes for index coding problems, generalizing clique and cycle covering on the side information digraph and further reducing the achievable broadcast rate. In this paper, we start with partition multicast, a special case of partial clique covering with cover number 1, and show that partition multicast achieves the optimal broadcast rate ...

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy Problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

A graph G is said to be a set graph if it admits an acyclic orientation that is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we continue the study of set graphs and related topics, focusing on computational complexity as...