an acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. the acyclic chromatic index of a graph $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

Let $p$ be prime and $alpha:x mapsto xg^x$, the Discrete Lambert Map. For $kgeq 1,$ let $ V = {0, 1, 2, . . . , p^k-1}$. The iteration digraph is a directed graph with $V$ as the vertex set and there is a unique directed edge from $u$ to $alpha(u)$ for each $uin V.$ We denote this digraph by $G(g, p^{k}),$ where $gin (mathbb{Z}/p^kmathbb{Z})^*.$  In this piece of work, we investigate the struct...

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to...

We study the complexity of deciding whether a given digraph D has a vertex-partition into two disjoint subdigraphs with given structural properties. Let H and E denote following two sets of natural properties of digraphs: H ={acyclic, complete, arcless, oriented (no 2-cycle), semicomplete, symmetric, tournament} and E ={strongly connected, connected, minimum outdegree at least 1, minimum in-deg...

Let D = (V, E) be a digraph with n vertices and m arcs. For each e E E there is an associated cost ce and a transit time te; Ce can be arbitrary, but we require t to be a non-negative integer. The cost to time ratio of a cycle C is X(C) = 3 ec ceCeec t. Let E' c E denote the set of arcs e with te > 0, let T = max{tv: (u, v) E} for each vertex u, and let T = uev T. We give a new algorithm for fi...

We study Pippenger’s model of Boolean networks with unreliable gates. In this model, the conditional probability that a particular gate fails, given the failure status of any subset of gates preceding it in the network, is bounded from above by some ε. We show that if a Boolean network with n gates is selected at random according to the Barak-Erdős model of a random acyclic digraph, such that t...

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D−S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016 ] considered the kernelization complexity of DFVS with an additional restriction on F , na...