Maximum directed cuts in acyclic digraphs

Abstract

It is easily shown that every digraph with m edges has a directed cut of size at least m/4, and that 1/4 cannot be replaced by any larger constant. We investigate the size of a largest directed cut in acyclic digraphs, and prove a number of related results concerning cuts in digraphs and acyclic digraphs.

Maximum directed cuts in digraphs with degree restriction

“…It was proved in [1] that every acyclic digraph of size m in D(1, 1) has a directed cut of at least 2m/5 edges. This is not true for all digraphs in D(1, 1).…”

“…Hence there are digraphs of size m with maximum directed cut not larger than m/3. On the other hand, it was shown in [1] that the edge set of every digraph D ∈ D(1, 1) has a decomposition into three directed cuts (see Theorem 7 below), hence D always contains a directed cut of size m/3.…”

“…Discussions in [1] show that a digraph D of size m has a cut of size )m. In [6], lower bounds for the largest directed cuts were asked for a family of digraphs with constrained indegree or outdegree. Let D(k, ℓ) be the family of all digraphs in which every vertex has either indegree at most k or outdegree at most ℓ (that is d…”

“…In Section 2 we consider the case k = ℓ = 1 and discuss the size of the maximum directed cut of digraphs in D(1, 1). It was proved in [1] that every acyclic digraph of size m in D(1, 1) has a directed cut of at least 2m/5 edges. From a result of Bondy and Locke [4] it is easy to see that the same lower bound holds for maximum directed cuts in triangle-free subcubic digraphs (a graph is subcubic if it has maximum degree at most three).…”

Parameterizing MAX SNP Problems Above Guaranteed Values

“…For such an optimization problem Q, the standard parameterized versionQ defined bỹ Q = {(I, k) : I is an instance of Q and opt(I) ≥ k} is easily seen to be fixed parameter tractable. For if k ≤ f (|I|), we answer 'yes'; else, f (|I|) < k and so |I| < f −1 (k) 1 and we have a kernel.…”