A graph G is said to be a set graph if it admits an acyclic orientation that is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we continue the study of set graphs and related topics, focusing on computational complexity as...

A digraph is semicomplete if any two vertices are connected by at least one arc and locally the out-neighbourhood in-neighbourhood of vertex induce a digraph. In this paper we study various subclasses digraphs for which give structural decomposition theorems. As consequence obtain several applications, among an answer to conjecture Naserasr first third authors (Aboulker et al., 2021): oriented ...

Let D be a digraph. The chromatic number χ(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or ∞ if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with χ(D) > k. It is shown that G(k, n) ≥ n1/k and G(k, n) ≤ (3 log2 n...

Consider a directed graph (digraph) in which vertices are assigned color sets, and two connected if only they share at least one the tail vertex has strictly smaller set than head. We seek to determine smallest possible size of union sets that allows for such digraph representation. To address this problem, we introduce new notion intersection representation digraph, show it is well-defined all...

We study hardness of approximating several minimaximal and maximinimal NP-optimization problems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted complete digraph. MINLOP is APX-hard but its unweighted version is polynomial time solvable. We prove that, MIN-MAX-SUBDAG problem, which is a generalization of MINLO...

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

In this paper we introduce a numerical invariant of digraphs which generalizes that of the number of connected components of a graph. The ao,clic disconnection ~(D) of a digraph D is the minimum number of (weakly) connected components of the subdigraphs obtained from D by deleting an acyclic set of arcs. We state some results about this invariant and compute its value for a variety of circulant...

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by CO(D) (CC(D)) and its size by co(D) (cc(D)). Gutin, Johnstone...

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orienta...

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.