Directed width parameters and circumference of digraphs

Abstract

We prove that the directed treewidth, DAG-width and Kelly-width of a digraph are bounded above by its circumference plus one.

Hitting Long Directed Cycles is Fixed-Parameter Tractable

“…The structure of long cycles in digraphs has been of interest for long time. For instance, Lewin [21] analyzed the density of such graphs, and Kintali [18] analyzes the directed treewidth of such directed graphs. Algorithmically, though, it was only recently shown by Kawarabayashi and Kreutzer [17] that the vertex version of the Erdős-Posa property holds for long directed cycles: namely, they show that any directed graph G either contains a set of k `1 vertex-disjoint directed cycles of length at least or some set S of at most f pk, q vertices that intersects all directed cycles of G with length at least .…”

“…Yet, for digraphs the k-linkage problem is NP-hard already for k " 2, and no fixed-parameter algorithm is known which recognizes digraphs of nearly-bounded directed treewidth. On the positive side, though, digraphs of bounded directed circumference are nicely squeezed between acyclic digraphs and digraphs of bounded directed treewidth [18]. Moreover, the arc version of the k-linkage problem is fixed-parameter tractable on digraphs of directed circumference 2 [3]; the question remains open for digraphs of arbitrary directed circumference.…”

DAG‐Width and Circumference of Digraphs

“…Kintali [14] proved that the DAG-width of a digraph is at most its circumference plus one and conjectured that the plus one could be removed. Using the game of cops and robbers, we are now ready to prove our main result which answers Kintali's conjecture in the affirmative.…”

“…D has circumference 4 and consist of six strong components. The vertex sets of the six strong components (listed according to an acyclic ordering) are S 1 = {1, 2}, S 2 = {27, 28}, S 3 = {5, 6}, S 4 = {3, 4}, S 5 = {7, 8,9,10,11,12,13,14,15,16,17, 18} and S 6 = {19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32}, and the strong component digraph SC(D) (obtained by contracting each strong component into a vertex) is Now for each strong component we use the strategy described in the proof of Theorem 3.1. Notice that S 1 , S 2 , S 3 and S 4 all consist of 2 vertices and hence here, in turn, we just place a cop on each vertex and after this we have chased the robber into either S 5 or S 6 .…”

“…Kintali [14] showed that digraphs with bounded circumference have DAG-width at most the circumference plus one and conjectured that the right bound is the circumference. We prove this conjecture in Section 3.…”