For a given $2$-partition $(V_1,V_2)$ of the vertices of a (di)graph $G$, we study properties of the spanning bipartite subdigraph $B_G(V_1,V_2)$ of $G$ induced by those arcs/edges that have one end in each $V_i$. We determine, for all pairs of non-negative integers $k_1,k_2$, the complexity of deciding whether $G$ has a 2-partition $(V_1,V_2)$ such that each vertex in $V_i$ has at least $k_i$ ...

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 D be a digraph on [Formula: see text]. Then the sequence [Formula: see text] is called the degree sequence of D. For any given sequence of pairs of integers [Formula: see text], if there exists a k-arc strongly connected digraph D such that d is the degree sequence of D, then d is realizable and D is a realization of d. In this paper, characterizations for k-arc-connected realizable sequenc...

If D is a strongly connected digraph, then an arc set S of D is called a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D − V (D1) contains an arc. Recently, Volkmann [12] defined the restricted arc-connectivity λ(D) as the minimum cardinality over all restricted arc-cuts S. A strongly connected digraph D is called λconnected when λ(D) exists. Let k ≥ 2 be an i...

The central observation of this paper is that if ǫn random arcs are added to any n-node strongly connected digraph with bounded degree then the resulting graph has diameter O(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs...

Given a digraph D, consider the model where each vertex is always operational, but the edges are independently operational with probability p. The strongly connected reliability of D, scRel(D, p), is the probability that the spanning subgraph of D consisting of the operational edges is strongly connected. One can view strongly connected reliability as the probability that any vertex can send in...

An m-colored digraph is a digraph whose arcs are colored with m colors. A directed path is monochromatic when its arcs are colored alike. A set S ⊆ V (D) is a kernel by monochromatic paths whenever the two following conditions hold: 1. For any x, y ∈ S, x 6= y, there is no monochromatic directed path between them. 2. For each z ∈ (V (D)− S) there exists a zS-monochromatic directed path. In this...

An in-tournament is an oriented graph such that the in-neighborhood of every vertex induces a tournament. Recently, we have shown that every arc of a strongly connected tournament of order n is contained in a directed path of order r(n + 3)/21. This is no longer valid for strongly connected in-tournaments, because there exist examples containing an arc with the property that the longest directe...

We say a digraph G is a minor of a digraph H if G can be obtained from a subdigraph of H by repeatedly contracting a strongly-connected subdigraph to a vertex. Here, we show the class of all tournaments is a well-quasi-order under minor containment.

The energy of a digraph D with eigenvalues z1, z2, . . . , zn is defined as E(D) = n ∑ j=1 |Rzj |, where Rzj is the real part of the complex number zj . In this paper, we characterize some positive reals that cannot be the energy of a digraph. We also obtain a sharp lower bound for the energy of strongly connected digraphs.