Ends in digraphs

AbstractIn this paper a sort of end concept for directed graphs is introduced and examined. Two one-way infinite paths are called equivalent iff there are infinitely many pairwise disjoint paths joining them. An end of an undirected graph is an equivalence class with respect to this relation. For two one-way infinite directed paths U and V define: (a) U⩽V iff there are infinitely many pairwise disjoint directed paths from U to V; (b) U ∼ V iff U ⩽ V and V ⩽ U. The relation ⩽ is a quasiorder, and hence ∼ is an equivalence relation whose classes are called ends. Furthermore, ⩽ induces a partial order on the set of ends of a digraph. In the main section, necessary and sufficient conditions are presented for an abstract order to be representable by the end order of a digraph

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

On Complexity of Minimum Leaf Out-branching Problem

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable