The notion of a p-competition graph and the p-competition number of a graph were introduced by S. -R. Kim, T. A. McKee, F. R. McMorris, and F. S. Roberts as a generalization of a competition graph and the competition number of a graph, respectively. Let p be a positive integer. The p-competition graph Cp(D) of a digraph D = (V, A) is a (simple undirected) graph which has the same vertex set V a...

We call a boolean formula (in CNF) stable acyclic if the associated digraph is free of a certain type of cycles. We show that satisfiability and recognition are polynomial–time solvable problems for stable acyclic formulas. Further, we show that a minimal unsatisfiable formula is stable acyclic if and only if the number of clauses exceeds the number of variables exactly by one (this subclass of...

We will define a matrix A=A1 associated with an acyclic digraph 1, such that the coefficients of the characteristic polynomial of A enumerate the paths in 1 according to their length. Our result is actually an easy consequence of a more general theorem of Goulden and Jackson, but this special case seems never to have been explicitly noted before. We will give two simple proofs, the first of whi...

A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D, denoted by dst(D), is the minimum number of galaxies needed to cover A(D). In this paper, we show that dst(D) ≤ Δ(D) + 1 and that if D is acyclic then dst(D) ≤ Δ(D). These results are proved by considering the existence of spann...

A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work...

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints need to be satisfied. The (weak resp.) digraph constraint imposes that if (all incoming nodes of resp.) a node x belongs to the solution, then the latter comp...

The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the outdegree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic digraph as a connected digraph which is either acyclic or has exactly one oriented cycle whose removal disconn...

An acyclic digraph in which every vertex has indegree at most i and outdegree j is called an (i,j) for some positive integers j. Lee et al. (2017) studied the phylogeny graphs of (2,2) digraphs gave sufficient conditions necessary a to have chordal graph. Their work was motivated by problems related evidence propagation Bayesian network it useful know moral (phylogeny are theory). In this paper...

An acyclic USO on a hypercube is formed by directing its edges in such as way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modeled as pivoting in a unit hypercube of the same dimension with an abstract objective function, and vice versa. In such a way, Zadeh’s ’least entered rule’ and other h...