Acyclic Subgraphs of Planar Digraphs

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 there exists an acyclic set of size at least $(1 - 3/g)n$.Comment: 9 page

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

Structural Routability of n-Pairs Information Networks

Information does not generally behave like a conservative fluid flow in communication networks with multiple sources and sinks. However, it is often conceptually and practically useful to be able to associate separate data streams with each source-sink pair, with only routing and no coding performed at the network nodes. This raises the question of whether there is a nontrivial class of network topologies for which achievability is always equivalent to routability, for any combination of source signals and positive channel capacities. This chapter considers possibly cyclic, directed, errorless networks with n source-sink pairs and mutually independent source signals. The concept of downward dominance is introduced and it is shown that, if the network topology is downward dominated, then the achievability of a given combination of source signals and channel capacities implies the existence of a feasible multicommodity flow.Comment: The final publication is available at link.springer.com http://link.springer.com/chapter/10.1007/978-3-319-02150-8_