Strong Asymmetric Digraphs with Prescribed Interior and Annulus

The directed distance d(u,v) from u to v in a strong digraph D is the length of a shortest u-v path in D. The eccentricity e(v) of a vertex v in D is the directed distance from v to a vertex furthest from v in D. The center and periphery of a strong digraph are two well known subdigraphs induced by those vertices of minimum and maximum eccentricities, respectively. We introduce the interior and annulus of a digraph which are two induced subdigraphs involving the remaining vertices. Several results concerning the interior and annulus of a digraph are presented.     

c-Pancyclic Partial Ordering and （c-1）-Pan-Outpath Partial Ordering in Semicomplete Multipartite Digraphs

An outpath of a vertex v in a digraph is a path starting at v such that v dominates the end vertex of the path only if the end vertex also dominates v.First we show that letting D be a strongly connected semicomplete c-partite digraph (c≥3)1 and one of the partite sets of it consists of a single vertex, say v, then D has a c-pancyclic partial ordering from v, which generalizes a result about pancyclicity of multipartite tournaments obtained by Gutin in 1993.Then we prove that letting D be a strongly connected semicomplete c-partite digraph with c≥3 and letting v be a vertex of D,then Dhas a(c-1)-pan-outpath partly ordering from v.This result improves a theorem about outpaths in semicomplete multipartite digraphs obtained by Guo in 1999.     

On strong digraphs with a prescribed ultracenter

The (directed) distance from a vertex u to a vertex v in a strong digraph D is the length of a shortest u-v (directed) path in D. The eccentricity of a vertex v of D is the distance from v to a vertex furthest from v in D. The radius radD is the minimum eccentricity among the vertices of D and the diameter diamD is the maximum eccentricity. A central vertex is a vertex with eccentricity radD and the subdigraph induced by the central vertices is the center C(D). For a central vertex v in a strong digraph D with radD < diamD, the central distance c(v) of v is the greatest nonnegative integer n such that whenever d(v, x) n, then x is in C(D). The maximum central distance among the central vertices of D is the ultraradius uradD and the subdigraph induced by the central vertices with central distance uradD is the ultracenter UC(D). For a given digraph D, the problem of determining a strong digraph H with UC(H) = D and C(H) D is studied. This problem is also considered for digraphs that are asymmetric.     

Directed distance in digraphs: Centers and medians

The directed distance d_D(u, v) from a vertex u to a vertex v in a strong digraph D is the length of a shortest (directed) u - v path in D. The eccentricity of a vertex v in D is the directed distance from v to a vertex furthest from v. The distance of a vertex v in D is the sum of the directed distances from v to the vertices of D. The center C(D) of D is the subdigraph induced by those vertices of minimum eccentricity, while the median M(D) of D is the subdigraph induced by those vertices of minimum distance. It is shown that for every two asymmetric digraphs D₁ and D₂, there exists a strong asymmetric digraph H such that C(H) ? D₁ and M(H) ? D₂, and where the directed distance from C(H) to M(H) and from M(H) to C(H) can be arbitrarily prescribed. Furthermore, if K is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both D₁ and D₂, then there exists a strong asymmetric digraph F such that C(F) ? D₁, M(F) ? D₂ and C(F) ∩ M(F) ? K. © 1993 John Wiley & Sons, Inc.     

Quasi-transitive digraphs

A digraph is quasi-transitive if there is a complete adjacency between the inset and the outset of each vertex. Quasi-transitive digraphs are interseting because of their relation to comparability graphs. Specifically, a graph can be oriented as a quasi-transitive digraph if and only if it is a comparability graph. Quasi-transitive digraphs are also of interest as they share many nice properties of tournaments. Indeed, we show that every strongly connected quasi-transitive digraphs D on at least four vertices has two vertices v₁ and v₂ such that D – v_i is strongly connected for i = 1, 2. A result of tournaments on the existence of a pair of arc-disjoint in- and out-branchings rooted at the same vertex can also be extended to quasi-transitive digraphs. However, some properties of tournaments, like hamiltonicity, cannot be extended directly to quasi-transitive digraphs. Therefore we characterize those quasi-transitive digraphs which have a hamiltonian cycle, respectively a hamiltonian path. We show the existence of highly connected quasi-transitive digraphs D with a factor (a collection of disjoint cycles covering the vertex set of D), which have a cycle of every length 3 ≦ k ≦ |V(D)| ? 1 through every vertex and yet they are not hamiltonian. Finally we characterize pancyclic and vertex pancyclic quasi-transitive digraphs. © 1995, John Wiley & Sons, Inc.     

Second Neighborhood via First Neighborhood in Digraphs

Let D be a simple digraph without loops or digons. For any v ? V(D) v\in V(D) , the first out-neighborhood N⁺(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N⁺⁺(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that |N⁺⁺(v)| 3 g|N⁺(v)| |N^{++}(v)|\geq\gamma|N^+(v)| , where % = 0.657298... is the unique real root of the equation 2x³ + x² -1 = 0.     

Hypotraceable digraphs

A hypotraceable digraph is a digraph D = (V, E) which is not traceable, i.e., does not contain a (directed)Hamiltonian path, but for which D - v is traceable for all ve ∈ V. We prove that a hypotraceable digraph of order n exists iff n ≥ 7 and that for each k ≥ 3 there are infinitely many hypotraceable oriented graphs with a source and a sink and precisely k strong components. We also show that there are strongly connected hypotraceable oriented graphs and that there are hypotraceable digraphs with precisely two strong components one of which is a source or a sink. Finally, we prove that hypo-Hamiltonian and hypotraceable digraphs may contain large complete subdigraphs.     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

Niche numbers

An acyclic digraph D =(V U X, <) induces a finite graph G =(V, E) if for all v,v′ if and only if a vertex of D dominates v and v′, or is dominated by v and v′. The niche number of an induced G is the cardinality of a smallest X for which some D induces G. Many induced graphs have niche number 0, 1, or 2, but it has been an open problem as to whether they can have larger niche numbers. We prove that there are induced graphs with arbitrarily large niche numbers. Explicit examples of graphs with niche numbers 3 and 4 are given. The smallest example uses 11 vertices. We note also that every induced graph with fewer than 8 vertices has niche number 0, 1, or 2.     

Kernels by monochromatic paths in digraphs with covering number 2

We call the digraph D an k-colored digraph if the arcs of D are colored with k colors. A subdigraph H of D is called monochromatic if all of its arcs are colored alike. A set N⊆V(D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) for every pair of different vertices u,v∈N, there is no monochromatic directed path between them, and (ii) for every vertex x∈(V(D)?N), there is a vertex y∈N such that there is an xy-monochromatic directed path. In this paper, we prove that if D is an k-colored digraph that can be partitioned into two vertex-disjoint transitive tournaments such that every directed cycle of length 3,4 or 5 is monochromatic, then D has a kernel by monochromatic paths. This result gives a positive answer (for this family of digraphs) of the following question, which has motivated many results in monochromatic kernel theory: Is there a natural numberlsuch that if a digraphDisk-colored so that every directed cycle of length at mostlis monochromatic, thenDhas a kernel by monochromatic paths?     

A lower bound for the recognition of digraph properties

The complexity of a digraph property is the number of entries of the adjacency matrix which must be examined by a decision tree algorithm to recognize the property in the worst case, Aanderaa and Rosenberg conjectured that there is a constant such that every digraph property which is monotone (not destroyed by the deletion of edges) and nontrivial (holds for some but not all digraphs) has complexity at leastv ² wherev is the number of nodes in the digraph. This conjecture was proved by Rivest and Vuillemin and a lower bound ofv ²/4–o(v ²) was subsequently found by Kahn, Saks, and Sturtevant. Here we show a lower bound ofv ²/2–o(v ²). We also prove that a certain class of monotone, nontrivial bipartite digraph properties is evasive (requires that every entry in the adjacency matrix be examined in the worst case).     

Coefficients of ergodicity and the scrambling index

For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.     

Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d⁺(u),d^-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when

λ(u,v)=min{d⁺(u),d^-(v)}     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.     

On CCE graphs of doubly partial orders

Let D be a digraph. The competition-common enemy graph (CCE graph) of D has the same set of vertices as D and an edge between vertices u and v if and only if there are vertices w and x in D such that (w,u), (w,v), (u,x), and (v,x) are arcs of D. We call a graph a CCE graph if it is the CCE graph of some digraph. In this paper, we show that if the CCE graph of a doubly partial order does not contain C₄ as an induced subgraph, it is an interval graph. We also show that any interval graph together with enough isolated vertices is the CCE graph of some doubly partial order.     

The signed <Emphasis Type="Italic">k</Emphasis>-domination number of directed graphs

Let k ≥ 1 be an integer, and let D = (V; A) be a finite simple digraph, for which d _D ⁻ ≥ k − 1 for all v ɛ V. A function f: V → {−1; 1} is called a signed k-dominating function (SkDF) if f(N ⁻[v]) ≥ k for each vertex v ɛ V. The weight w(f) of f is defined by $ \sum\nolimits_{v \in V} {f(v)} $ \sum\nolimits_{v \in V} {f(v)} . The signed k-domination number for a digraph D is γ _kS (D) = min {w(f|f) is an SkDF of D. In this paper, we initiate the study of signed k-domination in digraphs. In particular, we present some sharp lower bounds for γ _kS (D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs and digraphs.     

On the 2-cyclic property in 2-regular digraphs

A digraph D is said to be k-cyclic if every k vertices of D lie in some directed cycle of D . We show that almost every 2-regular digraph is 2-cyclic.     

Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

-Decompositions of and D v ∪P where P is a 2-Regular Subgraph of D v

In this paper, we extend the study of C₄-decompositions of the complete graph with 2-regular leaves and paddings to directed versions. Mainly, we prove that if P is a vertex-disjoint union of directed cycles in a complete digraph D_v, then and D_v∪P can be decomposed into directed 4-cycles, respectively, if and only if v(v−1)−|E(P)|≡0(mod 4) and v(v−1)+|E(P)|≡0(mod 4) where |E(P)| denotes the number of directed edges of P, and v≥8.