Strong Digraph Research Articles

Strong digraph groups

Abstract A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

Symmetric cores and extremal size bound for supereulerian semicomplete bipartite digraphs

A digraph D is supereulerian if D has a spanning eulerian subdigraph, or equivalently, a spanning closed trail. The symmetric core J=J(D) of a digraph D is a spanning subdigraph of D with A(J) consisting of all symmetric arcs in D. Let k(D) denote the number of connected symmetric components of J. We investigate conditions to assure that D has a spanning trail, and find a family of well-characterized nonsupereulerian digraphs H such that a strong digraph D with k(D)≤3 has a spanning closed trail if and only if D∉H. The main results are applied to obtain a tight bound on the size of a semicomplete bipartite digraph to be supereulerian, with all the extremal digraphs characterized.

Scattering number of digraphs

The scattering number of a strong digraph D is defined by s(D)=max{ω(D−R)−|R|}, where R and ω(D−R) respectively represent a separator of D and the number of strong components of D−R. This concept is a natural extension of the scattering number of undirected graphs which has a strong background of applications in measuring the vulnerability of networks.In this paper, we get sharp upper and lower bounds of the scattering number of general strong digraphs. We also determine the precise values for scattering numbers of some Cartesian product digraphs.

Relating g-good-neighbor connectivity and g-good-neighbor diagnosability of strong digraph network

In 2020, Lin and Zhang proposed the concept of g-good-neighbor (respectively, g-good-in-neighbor, g-good-out-neighbor) connectivity of digraphs and their respective diagnosability under PMC model. In this paper, we establish some relationships between connectivity, diagnosability, g-good-neighbor connectivity and g-good-neighbor (respectively, g-good-in-neighbor, g-good-out-neighbor) diagnosability of strong digraph networks. As an application, we use these relationships to obtain g-good-out-neighbor and 1-good-neighbor (respectively, 1-good-in-neighbor) diagnosability of some well-known strong digraph networks.

(Strong) Proper vertex connection of some digraphs

The (strong) proper vertex connection number spvc→(Γ), (s)pvc-number for short, is denoted as the smallest cardinality of colors required to color the digraph Γ so that Γ is (strong) properly vertex connected. The pvc-number and the spvc-number are calculated for some unique classes of digraphs in this paper, along with some fundamental results on these parameters. It is known that the pvc-number is not exceeding 3 for any strong digraph. For digraphs with pvc-number not exceeding 2, we provide some sufficient conditions. Furthermore, we prove that the spvc-number is at most 3 for any minimal strongly connected digraph, but it can be arbitrarily large for some strong digraphs.

The Strong Menger Connectivity of the Directed k-Ary n-Cube

A strong digraph [Formula: see text] is strongly Menger (arc) connected if, for [Formula: see text], [Formula: see text] can reach [Formula: see text] by min[Formula: see text] internally (arc) disjoint-directed paths. A digraph [Formula: see text] is [Formula: see text](-arc)-fault-tolerant strongly Menger([Formula: see text]-(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text]. A digraph [Formula: see text] is [Formula: see text]-conditional (arc)-fault-tolerant strongly Menger ([Formula: see text]-C(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text] and [Formula: see text]. The directed [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] [Formula: see text] and [Formula: see text] is a digraph with vertex set [Formula: see text]. For two vertices [Formula: see text] and [Formula: see text], [Formula: see text] dominates [Formula: see text] if there exists an integer [Formula: see text], [Formula: see text], satisfying [Formula: see text]mod [Formula: see text] and [Formula: see text], when [Formula: see text]. In this paper, we show that [Formula: see text] [Formula: see text] is [Formula: see text]-AFTSM arc connected when [Formula: see text], [Formula: see text]-FTSM connected when [Formula: see text], [Formula: see text]-CAFTSM arc connected when [Formula: see text], and [Formula: see text]-CFTSM connected when [Formula: see text].

Maximum size of digraphs of given radius

In 1967, Vizing determined the maximum size of a graph with given order and radius. In 1973, Fridman answered the same question for digraphs with given order and outradius. We investigate that question when restricting to strong digraphs. Strong digraphs are the digraphs with a finite total distance and hence the interesting ones, as we want to note a connection between minimizing the total distance and maximizing the size under the same constraints. We characterize the extremal digraphs maximizing the size among all strong digraphs of order n and outradius 3, as well as when the order is sufficiently large compared to the outradius. As such, we solve a problem of Dankelmann asymptotically. We also consider these questions for bipartite digraphs and solve a second problem of Dankelmann partially. The full paper can also be found at arxiv.org/abs/2201.00186.

4-Free Strong Digraphs with the Maximum Size

Directed cycles in digraphs are useful in embedding linear arrays and rings, and are suitable for designing simple algorithm with low communication costs in parallel computer systems, thus the existence of directed cycles on digraphs has been largely investigated. Let [Formula: see text], [Formula: see text] be integers. Bermond et al. [Journal of Graph Theory 4(3) (1980) 337–341] proved that if the size of a strong digraph [Formula: see text] with order [Formula: see text] is at least [Formula: see text], then the girth of [Formula: see text] is no more than [Formula: see text]. Consequently, when [Formula: see text] is a 4-free strong digraph with order [Formula: see text], which means that every directed cycle in [Formula: see text] has length at least [Formula: see text], then the maximum size of [Formula: see text] is [Formula: see text]. In this paper, we mainly give the structural characterizations for all 4-free strong digraphs of order [Formula: see text] whose arc number exactly is [Formula: see text].

Subdivisions of oriented cycles in Hamiltonian digraphs with small chromatic number

Cohen et al. conjectured that, for each oriented cycle C, there exists an integer f(C) such that any f(C)-chromatic strong digraph contains a subdivision of C as a subdigraph. In the same paper, Cohen et al. proved this conjecture for cycles with two blocks by showing that the chromatic number of strong digraphs that include no subdivision of a cycle with two blocks, with lengths of k1 and k2, is bounded from above by O((k1+k2)4). More recently, Kim et al. improved this upper bound to O((k1+k2)2) for the class of strong digraphs and to O(k1+k2) for the class of Hamiltonian digraphs. In this paper, we confirm Cohen et al.'s conjecture for Hamiltonian digraphs. We demonstrate a stronger version by showing that every 3n-chromatic Hamiltonian digraph contains a subdivision for every oriented cycle of order n.

Turán number of 3-free strong digraphs with out-degree restriction

A digraph D is r-free if such D has no directed cycles of length at most r, where r is a positive integer. In 1980, Bermond et al. showed that if D is an r-free strong digraph of order n, then the size of D is at most n−r+22+r−2 and the upper bound is tight, for r≥2. Namely, they determined the Turán number of Cr-free strong digraphs of order n, where Cr={C2,C3,…,Cr} and Ci is a directed cycle of length i∈{2,3,…,r}. Specially, for r=3, the maximum size of 3-free strong digraphs of order n is n−12+1. Let Φn(ξ,γ) be a family of 3-free strong digraphs of order n in which the minimum out-degree is at least ξ and the minimum in-degree is at least γ, where both ξ and γ are positive integers. Let φn(ξ,γ) be the maximum size of digraphs of Φn(ξ,γ), i.e., Turán number of 3-free strong digraphs of order n with out and in-degree restrictions. We denote ϕn(ξ,γ)={D∈Φn(ξ,γ):thesizeofDisequaltoφn(ξ,γ)}. Recently, Chen et al. described ϕn(1,1), i.e., all 3-free strong digraphs of order n with size n−12+1. They also gave the bound of φn(2,1), that is, n−12−2≤φn(2,1)≤n−12. In this paper, we improve the upper bound of φn(2,1) to n−12−1 and we thus get n−12−2≤φn(2,1)≤n−12−1. In addition, we show that φn(2,1)=n−12−1 by constructing two 3-free strong digraphs with minimum out-degree two whose size reaches n−12−1 for n=7,8, and verify φn(2,1)=n−12−2 for n=9. As a consequence, Turán number of 3-free strong digraphs of order n with out-degree at least two, i.e., φn(2,1), is one of n−12−1 and n−12−2.

Arc‐disjoint in‐ and out‐branchings in digraphs of independence number at most 2

AbstractWe prove that every digraph of independence number at most 2 and arc‐connectivity at least 2 has an out‐branching and an in‐branching which are arc‐disjoint (we call such branchings a good pair). This is best possible in terms of the arc‐connectivity as there are infinitely many strong digraphs with independence number 2 and arbitrarily high minimum in‐ and out‐degrees that have no good pair. The result settles a conjecture by Thomassen for digraphs of independence number 2. We prove that every digraph on at most 6 vertices and arc‐connectivity at least 2 has a good pair and give an example of a 2‐arc‐strong digraph on 10 vertices with independence number 4 that has no good pair. We also show that there are infinitely many digraphs with independence number 7 and arc‐connectivity 2 that have no good pair. Finally we pose a number of open problems.

Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties

AbstractGeneralizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity of . In particular, together with a well‐known result due to Nash‐Williams and Tutte, this implies that every 7‐edge‐connected graph contains a spanning bipartite graph whose edge set decomposes into two edge‐disjoint spanning trees. We show that this is best possible as it does not hold for infinitely many 6‐edge‐connected graphs. For directed graphs, it was shown by Bang‐Jensen et al. that there is no such that every ‐arc‐connected digraph has a spanning strong bipartite subdigraph. We prove that every strong digraph has a spanning strong 3‐partite subdigraph and that every strong semicomplete digraph on at least six vertices contains a spanning strong bipartite subdigraph. We generalize this result to higher connectivities by proving that, for every positive integer , every ‐arc‐connected digraph contains a spanning ()‐partite subdigraph which is ‐arc‐connected and this is best possible. A conjecture by Kreutzer et al. implies that every digraph of minimum out‐degree contains a spanning 3‐partite subdigraph with minimum out‐degree at least . We prove that the bound would be best possible by providing an infinite class of digraphs with minimum out‐degree which do not contain any spanning 3‐partite subdigraph in which all out‐degrees are at least . We also prove that every digraph of minimum semidegree at least contains a spanning 6‐partite subdigraph in which every vertex has in‐ and out‐degree at least .

Matching and spanning trails in digraphs

Let D be a digraph and let α(D), α′(D) and λ(D) be independence number, the matching number and the arc-strong connectivity of D, respectively. Bang-Jensen and Thommassé in 2011 conjectured that every digraph D with λ(D)≥α(D) is supereulerian. In [J. Graph Theory, 81(4), (2016) 393-402], it is shown that every digraph D with λ(D)≥α′(D) is supereulerian. In this paper, we introduced the symmetric core of a digraph and use it to show that each of the following holds for a strong digraph D on n≥3 vertices with λ(D)≥α′(D)−1.(i) There exists a family D(n) of well-characterized digraphs such that for any digraph D with α′(D)≤2, D has a spanning trial if and only if D is not a member in D(n).(ii) If α′(D)≥3, then D has a spanning trail.(iii) If α′(D)≥3 and n≥2α′(D)+3, then D is supereulerian.(iv) If λ(D)≥α′(D)≥4 and n≥2α′(D)+3, then for any pair of vertices u and v of D, D contains a spanning (u,v)-trail.

Symmetric core and spanning trails in directed networks

A digraph D is supereulerian if D has a spanning closed trail, and is strongly trail-connected if for any pair of vertices u,v∈V(D), D has a spanning (u,v)-trail and a spanning (v,u)-trail. The symmetric core J=J(D) of a digraph D is a spanning subdigraph of D with A(J) consisting of all symmetric arcs in D. Let J1,J2,⋯,Jk(D) be the connected symmetric components of J and define λ0(D)=min⁡{λ(Ji):1≤i≤k(D)}. We prove that the contraction D′=D/J can be used to predict the existence of spanning trails in D. It is known that if k(D)≤2, then D has a spanning closed trail. In particular, each of the following holds for a strong digraph D with k(D)≥3.(i) If λ0(D)≥k(D)−2, then D has a spanning trail if and only if D′ has a spanning trail.(ii) If λ0(D)≥k(D)−1, then D is supereulerian if and only if D′ is supereulerian.(iii) If λ0(D)≥k(D), then D is strongly trail-connected if and only if D′ is strongly trail-connected.

Neighborhood Conditions for Super-λ′ Digraphs

Restricted arc-connectivity is a more refined network reliability index than arc-connectivity. An arc subset [Formula: see text] of a strong digraph [Formula: see text] is a restricted arc-cut of [Formula: see text] if [Formula: see text] has a non-trivial strong component [Formula: see text] such that [Formula: see text] contains an arc. The restricted arc-connectivity is the minimum cardinality over all restricted arc-cuts of [Formula: see text]. A strong digraph is super-[Formula: see text] if its restricted arc-connectivity is maximal and the number of its minimum restricted arc-cuts is minimal. In this paper, we present some neighborhood conditions for a digraph to be super-[Formula: see text].

3-Free Strong Digraphs with the Maximum Size

3-Free Strong Digraphs with the Maximum Size

Sufficient Ore type condition for a digraph to be supereulerian

A digraph D is called supereulerian if D has a spanning eulerian subdigraph. In this article, we show that a strong digraph D with n vertices is supereulerian if for every three different vertices z,w and v such that z and w are nonadjacent, d(z)+d(w)+d+(z)+d−(v)≥3n−5 (if (z,v)∉A(D)) and d(z)+d(w)+d−(z)+d+(v)≥3n−5 (if (v,z)∉A(D)). And this bound is sharp.

Digraphs with proper connection number two

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc→(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc→(D)=1 if and only if D is complete. Magnant et al. showed that pc→(D)≤3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).

A study on distance k-domination in digraphs

Let D = (V, A) be a finite and simple directed graph (digraph) with vertex set V and arc set A. For an integer k ≥ 1, the out k-neighborhood of a vertex ν is defined as . A set S ⊆ V is said to be a distance k-dominating set in D if N+ k (S) ∪ S = V. The minimum distance k-dominating set in D is the distance k-domination number, . For two integers s ≥ 2 and k, the (s, k) - kernal set and the (s, k) - kernal number, are studied. This parameter is also computed for different types of digraphs. Strong digraphs (SDs) are studied and of SDs are determined in terms of the radius and diameter of SDs. Also (2, k) - kernals of directed wounded spider are classified with . Some upper and lower bounds on are determined and characterized some digraphs achieving these bounds.

<i>Supereulerian Digraph</i> Strong Products

A vertex cycle cover of a digraph H is a collection C = {C1, C2, …, Ck} of directed cycles in H such that these directed cycles together cover all vertices in H and such that the arc sets of these directed cycles induce a connected subdigraph of H. A subdigraph F of a digraph D is a circulation if for every vertex in F, the indegree of v equals its out degree, and a spanning circulation if F is a cycle factor. Define f (D) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from D by contracting all arcs in F, among all circulations F of D. Adigraph D is supereulerian if D has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C’ with |C’| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. In this paper, we prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f (D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian.