Arc-Disjoint Paths and Trees in 2-Regular Digraphs

An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: 1) Given a 2-regular digraph $D$, decide if it contains two arc-disjoint branchings B^+_u, B^-_v. 2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u. Both problems are NP-complete for general digraphs. We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. We also prove that, for 2-regular digraphs, the latter problem is in fact equivalent to deciding if $D$ contains two arc-disjoint out-branchings. We generalize this result to k-regular digraphs where we want to find a number of pairwise arc-disjoint spanning trees and out-branchings such that there are k in total, again without prescribing any roots.Comment: 9 pages, 7 figure

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ 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

On DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure

Hereditarily hard H-colouring problems

AbstractLet H be a graph (respectively digraph) whose vertices are called ‘colours’. An H-colouring of a graph (respectively digraph) G is an assignment of these colours to the vertices of G so that if u is adjacent to v in G, then the colour of u is adjacent to the colour of v in H. We continue the study of the complexity of the H-colouring problem ‘Does a given graph (respectively digraph) admit an H-colouring?’. For graphs it was proved that the H-colouring problem is NP-complete whenever H contains an odd cycle, and is polynomial for bipartite graphs. For directed graphs the situation is quite different, as the addition of an edge to H can result in the complexity of the H-colouring problem shifting from NP-complete to polynomial. In fact, there is not even a plausible conjecture as to what makes directed H-colouring problems difficult in general. Some order may perhaps be found for those digraphs H in which each vertex has positive in-degree and positive out-degree. In any event, there is at least, in this case, a conjecture of a classification by complexity of these directed H-colouring problems. Another way, which we propose here, to bring some order to the situation is to restrict our attention to those digraphs H which, like odd cycles in the case of graphs, are hereditarily hard, i.e., are such that the H′-colouring problem is NP-hard for any digraph H′ containing H as a subdigraph. After establishing some properties of the digraphs in this class, we make a conjecture as to precisely which digraphs are hereditarily hard. Surprisingly, this conjecture turns out to be equivalent to the one mentioned earlier. We describe several infinite families of hereditarily hard digraphs, and identify a family of digraphs which are minimal in the sense that it would be sufficient to verify the conjecture for members of that family