We provide the first non-trivial approximation algorithm for the fundamental directed feedback vertex set (DFVS) problem in the class of quasi-transitive digraphs. This class of digraphs encompasses both dense and sparse classes of digraphs, for which specialized DFVS algorithms were proposed in the literature, like tournaments or transitive orientations of bounded treewidth graphs. Our approximation algorithm can handle both dense graphs, as well as sparse graphs, by a single approach, which is based on carefully analysing the solutions to a linear programming relaxation of DFVS. It also handles the node-weighted DFVS problem, for which it computes a 9/4-approximation in polynomial time. Along the way, we improve and simplify the best-known deterministic polynomial-time approximation algorithms for DFVS in tournaments (Cai et al., SICOMP 2001; Mnich et al., ESA 2016).

Abstract. We prove that for every set [Formula: see text] of vertices of a directed graph [Formula: see text], the maximum number of vertices in [Formula: see text] contained in a collection of vertex-disjoint cycles in [Formula: see text] is at least the minimum size of a set of vertices that hits all cycles containing a vertex of [Formula: see text]. As a consequence, the directed tree-width of a directed graph is linearly bounded in its cycle-width, which improves the previously known quadratic upper bound. We further show that the corresponding statement in bidirected graphs is true and that its edge-variant holds in both undirected and directed graphs, but fails in bidirected graphs. The vertex-version in undirected graphs remains an open problem.

Abstract. In 1996 Guiduli and Mohar proposed a conjecture that predicts the structure of connected graphs with minimum degree [Formula: see text] and minimum algebraic connectivity. We settle this conjecture for the case [Formula: see text]. As a result, we conclude that the minimum algebraic connectivity of connected graphs with [Formula: see text] vertices and [Formula: see text] is [Formula: see text], where [Formula: see text] is a function in [Formula: see text] that tends to 0 as [Formula: see text] goes to infinity. This enables us to provide a positive answer to the problem of whether graphs with [Formula: see text] and nearly maximum diameter have asymptotically minimum algebraic connectivity.