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...

Let D be a finite directed acyclic multigraph and t be a vertex of D such that for each other vertex x of D, there are n pairwise openly disjoint paths in D from x to t. It is proved that there exist n spanning trees BI , , B,, in D directed toward t such that for each vertex x # t of D, the N paths from x to t in B1, ,B,, are pairwise openly disjoint. @ 1999 Elsevier Science B.V. All rights re...

In k-Digraph Coloring we are given a digraph and asked to partition its vertices into at most k sets, so that each set induces DAG. This well-known problem is NP-hard, as it generalizes (undirected) k-Coloring, but becomes trivial if the input acyclic. poses natural parameterized complexity question of what happens when “almost” this paper study using parameters measure input’s distance acyclic...

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, implies the existence of the (non-homogeneous) cd-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian po...

The dichromatic number of D, denoted by χ→(D), is the smallest integer k such that D admits an acyclic k-coloring. We use maderχ→(F) to denote if χ→(D)≥k, then contains a subdivision F. A digraph F called Mader-perfect for every subdigraph F′ F, maderχ→(F′)=|V(F′)|. extend octi digraphs larger class and prove it Mader-perfect, which generalizes result Gishboliner, Steiner Szabó [Dichromatic for...

‎let $g$ be a graph and $chi^{prime}_{aa}(g)$ denotes the minimum number of colors required for an‎ ‎acyclic edge coloring of $g$ in which no two adjacent vertices are incident to edges colored with the same set of colors‎. ‎we prove a general bound for $chi^{prime}_{aa}(gsquare h)$ for any two graphs $g$ and $h$‎. ‎we also determine‎ ‎exact value of this parameter for the cartesian product of ...

A set X of vertices of an acyclic digraph D is convex if X ̸= ∅ and there is no directed path between vertices of X which contains a vertex not in X. A set X is connected if X ̸= ∅ and the underlying undirected graph of the subgraph ofD induced byX is connected. Connected convex sets and convex sets of acyclic digraphs are of interest in the area of modern embedded processor technology. We constr...

We consider the problem Maximum Leaf Spanning Tree (MLST) on digraphs, which is defined as follows. Given a digraph G, find a directed spanning tree of G that maximizes the number of leaves. MLST is NP-hard. Existing approximation algorithms for MLST have ratios of O( √ OPT) and 92. We focus on the special case of acyclic digraphs and propose two lineartime approximation algorithms; one with ra...

In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a perfect graph. However, an open question is the complexity status of the computation of a kernel in such a digraph. Our main contribution is to prove new polyn...

Competition graphs have appeared in a variety of applications from food webs to communication networks to energy models. The competition graph, C (D), of a digraph D has the same vertex set as D and (x; y) is an edge in C (D) if and only if there is a vertex z such that (x; z) and (y; z) are arcs in D. It has been observed that most actual food webs have interval competition graphs. Subsequent ...