Strong Digraph Research Articles

Lower and upper orientable strong radius and strong diameter of complete k-partite graphs

For two vertices u and v in a strong digraph D , the strong distance sd ( u , v ) between u and v is the minimum size (the number of arcs) of a strong sub-digraph of D containing u and v . For a vertex v of D , the strong eccentricity se ( v ) is the strong distance between v and a vertex farthest from v . The strong radius srad ( D ) (resp. strong diameter sdiam ( D ) ) is the minimum (resp. maximum) strong eccentricity among the vertices of D . The lower (resp. upper) orientable strong radius srad ( G ) (resp. SRAD ( G ) ) of a graph G is the minimum (resp. maximum) strong radius over all strong orientations of G. The lower (resp. upper) orientable strong diameter sdiam ( G ) (resp. SDIAM ( G ) ) of a graph G is the minimum (resp. maximum) strong diameter over all strong orientations of G. In this paper, we determine the lower orientable strong radius and diameter of complete k -partite graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of complete k -partite graphs. We also find an error about the lower orientable strong diameter of complete bipartite graph K m , n given in [Y.-L. Lai, F.-H. Chiang, C.-H. Lin, T.-C. Yu, Strong distance of complete bipartite graphs, The 19th Workshop on Combinatorial Mathematics and Computation Theory, 2002, pp. 12–16], and give a rigorous proof of a revised conclusion about sdiam ( K m , n ) .

On the number of noncritical vertices in strongly connected digraphs

By definition, a vertex w of a strongly connected (or, simply, strong) digraph D is noncritical if the subgraph D — w is also strongly connected. We prove that if the minimal out (or in) degree k of D is at least 2, then there are at least k noncritical vertices in D. In contrast to the case of undirected graphs, this bound cannot be sharpened, for a given k, even for digraphs of large order. Moreover, we show that if the valency of any vertex of a strong digraph of order n is at least 3/4n, then it contains at least two noncritical vertices. The proof makes use of the results of the theory of maximal proper strong subgraphs established by Mader and developed by the present author. We also construct a counterpart of this theory for biconnected (undirected) graphs.

Minimum k‐geodetically connected digraphs

AbstractA strong digraph is k‐geodetically connected (k‐GC) if the removal of at least k vertices is required to increase the distance between at least one pair of vertices or reduce it to a single vertex. Such digraphs can serve as models of distance invulnerable networks (immune to k – 1 or fewer vertex failures) for a system with one‐way communications. For every integer n, we determine the minimum size 2‐GC digraphs of order n. Further, we find the minimum size of a k‐GC digraph of order n if n ≡ 0 (mod k) and give good bounds for all n. Also, several operations on and constructions of k‐GC digraphs are presented. We show that the problem of finding a minimum size k‐GC spanning subdigraph of a given digraph is NP‐hard. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 243–253 2004

Minimum average distance of strong orientations of graphs

The average distance of a graph (strong digraph) G, denoted by μ( G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then μ → min (G) is the minimum average distance taken over all strong orientations of G. A lower bound for μ → min (G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for μ → min (G) in terms of μ( G). However, if every edge of G lies on 3-cycle, then it is shown that μ → min (G)⩽ 7 4 μ(G) . This bound is improved for maximal planar graphs to 5 3 μ(G) and even further to 3 2 μ(G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.

On the number of quasi‐kernels in digraphs

AbstractA vertex set X of a digraph D = (V, A) is a kernel if X is independent (i.e., all pairs of distinct vertices of X are non‐adjacent) and for every v ∈ V‐X there exists x ∈ X such that vx ∈ A. A vertex set X of a digraph D = (V, A), is a quasi‐kernel if X is independent and for every v ∈ V‐X there exist w ∈ V‐X, x ∈ X such that either vx ∈ A or vw, wx ∈ A. In 1974, Chvátal and Lovász proved that every digraph has a quasi‐kernel. In 1996, Jacob and Meyniel proved that if a digraph D has no kernel, then D contains at least three quasi‐kernels. We characterize digraphs with exactly one and two quasi‐kernels, and, thus, provide necessary and sufficient conditions for a digraph to have at least three quasi‐kernels. In particular, we prove that every strong digraph of order at least three, which is not a 4‐cycle, has at least three quasi‐kernels. © 2004 Wiley Periodicals, Inc.

The Directed Geodetic Structure of a Strong Digraph

By a ternary structure we mean an ordered pair (U 0, T 0), where U 0is a finitenonempty set and T 0is a ternary relation on U 0. A ternary structure (U 0, T 0) is called here a directed geodetic structure if there exists a strong digraph Dwith the properties that V(D) = U 0and T 0 (u,v, w)if and only if d D (u,v)+ d D (v,w)= d D (u, w) for all u, v, w∈ U 0, where d Ddenotes the (directed) distance function in D. It is proved in this paper that there exists no sentence sof the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies s.

Steiner type problems for digraphs that are locally semicomplete or extended semicomplete

AbstractWe consider the following three problems: (P1) Let D be a strong digraph and let X be a non‐empty subset of its vertices. Find a strong subdigraph D′ of D which contains all vertices of X and has as few arcs as possible. This problem is also known under the name the directed Steiner problem. (P2) Let D be a strong digraph with real‐valued costs on the vertices. Find a strong subdigraph of D of minimum cost. (P3) Let D be a strong digraph with real‐valued costs on the vertices. Find a cycle of minimum cost in D. All three problems are NP‐hard for general digraphs. The second problem generalizes the problem of finding a smallest strong subdigraph (measured in the number of vertices) which covers a given non‐empty set X of vertices. We describe polynomial algorithms for the problems (P1) and (P2) in the case when D is either locally semicomplete or extended semicomplete and for (P3) in the case of locally semicomplete digraphs. A polynomial algorithm for (P3) in the case of extended semicomplete digraphs was given by Bang‐Jensen et al., 2002. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 193–207, 2003

Every strong digraph has a spanning strong subgraph with at most n+2 α−2 arcs

Answering a question of Adrian Bondy (Per. Comm.), we prove that every strong digraph has a spanning strong subgraph with at most n+2 α−2 arcs, where α is the size of a maximum stable set of D. Such a spanning subgraph can be found in polynomial time. An infinite family of oriented graphs for which this bound is sharp was given by Odile Favaron (Discrete Math. 146 (1995) 289). A direct corollary of our result is that there exists 2 α−1 directed cycles which span D. Tibor Gallai (Theory of Graphs and its Applications, Czech. Acad. Sci. Prague, 1964, p. 161) conjectured that α directed cycles would be enough.

Orientations of digraphs almost preserving diameter

An orientation of a digraph D is a spanning subdigraph of D obtained from D by deleting exactly one arc between x and y for every pair x≠ y of vertices such that both xy and yx are in D. In this paper, we consider certain well-known classes of strong digraphs, each member D of which has an orientation with diameter not exceeding the diameter of D by more than a small constant.

On $k$-strong distance in strong digraphs

On $k$-strong distance in strong digraphs

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.

Partitioning Vertices of a Tournament into Independent Cycles

Let k be a positive integer. A strong digraph G is termed k-connected if the removal of any set of fewer than k vertices results in a strongly connected digraph. The purpose of this paper is to show that every k-connected tournament with at least 8k vertices contains k vertex-disjoint directed cycles spanning the vertex set. This result answers a question posed by Bollobás.

Distance in Digraphs

The (directed) distance d⃗( u, v) from a vertex u to a vertex v in a strong digraph D is the length of a shortest u-v path in D. Although this is the standard distance in digraphs, it is not a metric. Two other distances in digraphs are introduced, each of which is a metric. The maximum distance md( u, v) between two vertices u and v in a strong digraph is defined as md( u, v) = max{ d⃗( u, v), d⃗( v, u)}. The sum distance sd( u, v) is defined as sd( u, v) = d⃗( u, v) + d⃗( v, u). Several results and problems concerning these metrics and such parameters as centers, medians, and peripheries are described.

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.

Parallel algorithms for the hamiltonian cycle and hamiltonian path problems in semicomplete bipartite digraphs

We give anO(log4n)-timeO(n2)-processor CRCW PRAM algorithm to find a hamiltonian cycle in a strong semicomplete bipartite digraph,B, provided that a factor ofB (i.e., a collection of vertex disjoint cycles covering the vertex set ofB) is computed in a preprocessing step. The factor is found (if it exists) using a bipartite matching algorithm, hence placing the whole algorithm in the class Random-NC.

On minimum locally $n$-(arc)-strong digraphs

On minimum locally $n$-(arc)-strong digraphs

A short proof of the Chen-Manalastas theorem

Gallai and Milgram (1960) proved that a digraph with stability number α is spanned by α disjoint directed paths. Chen and Manalastas Jr (1983) proved that a strong digraph with stability number at most two is spanned by at most two consistent directed circuits. We slightly simplify the proof of the Gallai-Milgram theorem, while at the same time refining its statement, and use this sharpened version to obtain a relatively short proof of the Chen-Manalastas theorem. We also give a counterexample to a generalization of the Gallai-Milgram theorem conjectured by Hartman (1988).

Directed distance in digraphs: Centers and medians

AbstractThe directed distance dD(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 &amp; Sons, Inc.

A note on the exponent set of primitive minimally strong digraphs

We obtain a lower bound for e( n), the least integer (greater than or equal to 5) that is not the exponent of any nX n primitive, nearly reducible matrix. The main result is that under certain hypotheses about the distance between n and the nearest prime number we have that e( n)> n 2⧸3.

Covering the vertices of a digraph by cycles of prescribed length

Let D be a strong digraph with n vertices and at least ( n − 1)( n − 2) + 3 arcs. For any integers k, n 1, n 2,…, n k such that n = n 1 + n 2 +⋯+ n k and n i ⩾3, there exis covering of the vertices of D by disjoint directed cycles of length n 1, n 2,…, n k except in two cases: 1. Case 1: n = 6; n 1 = n 2 = 3 and D contains a stable set with 3 vertices. 2. Case 2: n = 9; n 1 = n 2 = n 3 = 3 and D contains a stable set with 4 vertices.