We show that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. We bound the number of edges needed for such an extension, and we apply the characterization to a problem on non-transitive dice.

Let D = (V,A) be a directed graph. For U ⊆ V , δin(U) and δout(U) denote the sets of arcs entering U and leaving U , respectively. Moreover, δ(U) := δin(U) ∪ δout(U), din(U) := |δin(U)| (the indegree), dout(U) := |δout(U)| (the outdegree), and d(U) := |δ(U)| (the degree or total degree). If U = {u} is a singleton, we replace the argument {u} by u. We attach subscript D or A if useful. For B ⊆ A...

Jirasek, J., Some remarks on Adam’s conjecture for simple directed graphs, Discrete Mathematics 108 (1992) 327-332. The classes of multidigraphs for which Adam’s conjecture (that any digraph containing a directed cycle has an arc whose reversal decreases the total number of directed cycles) does not hold, were described in Grinberg (1988), Jirasek (1987) and Thomassen (1987). The question remai...

In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree with at most k vertices of reduced out-degree. This problem is a directed analog of the wellstudied Minimum-Vertex Feedback Edge Set problem. We show that this problem is fixed-parameter tractable and admits a...

A connected graph with edge connectivity λ and minimum degree δ is called maximally edge-connected if λ = δ. A strongly connected digraph with arc connectivity λ and minimum degree δ is called maximally arc-connected if λ = δ. In this paper, some new sufficient conditions are presented for maximally edge-connected graphs and arc-connected digraphs. We only consider finite graphs (digraphs) with...

In this paper, we study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for three tournaments as well as extremal families for each tournament. We use Razborov’s semidefinite method for flag algebras to prove the upper bounds.

In argumentation theory, Dung’s abstract framework provides a unifying view of several alternative semantics based on the notion of extension. Recently, a new semantics has been introduced to solve the problems related to counterintuitive results produced by literature proposals. In this semantics, extensions can be decomposed and constructed along the strongly connected components of the defea...

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we characterize the antichain cutsets in semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We ...

Article history: Received 1 October 2012 Received in revised form 9 February 2015 Accepted 10 February 2015 Available online 14 February 2015

The existence of relational products in categories of relations is strongly connected with the representability of that category. In this paper we propose a canonical weakening of the notion of a relational product. Unlike the strong version any (small) category of relations can be embedded into a suitable category providing all weak relational products. Furthermore, we investigate the categori...