Strong Digraph Research Articles

Proximity and remoteness in directed and undirected graphs

Let D be a strongly connected digraph. The average distance σ̄(v) of a vertex v of D is the arithmetic mean of the distances from v to all other vertices of D. The remoteness ρ(D) and proximity π(D) of D are the maximum and the minimum of the average distances of the vertices of D, respectively. We obtain sharp upper and lower bounds on π(D) and ρ(D) as a function of the order n of D and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament T, we have π(T)=ρ(T) if and only if T is regular. Due to this result, one may conjecture that every strong digraph D with π(D)=ρ(D) is regular. We present an infinite family of non-regular strong digraphs D such that π(D)=ρ(D). We describe such a family for undirected graphs as well.

Counting acyclic and strong digraphs by descents

A descent of a labeled digraph is a directed edge (s,t) with s>t. We count strong tournaments, strong digraphs, acyclic digraphs, and forests by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.

Some results on the existence of Hamiltonian cycles in -compositions of bipartite digraphs

Let D be a digraph on n vertices s1, …, sn and let D1, …, Dn be a family of vertex-disjoint bipartite digraphs. We think of D1, …, Dn as 2-colored digraphs with the same color set. The -composition …, is the digraph obtained from D by replacing each vertex si of D by Di and adding an arc from each vertex of Di to each vertex of Dj if and only if those vertices have different color and is an arc of D (with i, and ). Notice that this is a generalization of the usual composition D, whenever each of the digraphs D1, …, Dn are k-partite, we obtain a k-partite digraph as a -composition and this digraph is a subdigraph of the usual composition …, In our case, we obtain a bipartite digraph. Particularly, the -composition …, when is the 2- or 3-cycle and each Di is a semicomplete bipartite digraph, is also a semicomplete bipartite digraph. Gutin, Häggkvist and Manoussakis proved that a semicomplete bipartite digraph has a Hamiltonian cycle if and only if it is strong and has a cycle-factor. In this article, we prove that a digraph …, where is the n-cycle and each Di is a strong semicomplete bipartite digraph, has a Hamiltonian cycle if and only if it is strong and has a cycle-factor.

Vertex-monochromatic connectivity of strong digraphs

A vertex coloring of a strong digraph D is a strong vertex-monochromatic connection coloring (SVMC-coloring) if for every pair u,v of vertices in D there exists an (u,v)-path having all its internal vertices of the same color. Let smcv(D) denote the maximum number of colors used in an SVMC-coloring of a digraph D. In this paper we determine the value of smcv(D), whenever D is the line digraph of a digraph. Also, if T is a tournament, we give conditions to find the exact value of smcv(T).

Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r. Let T be a digraph with t vertices u1,…,ut and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji∣1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp∣uiup∈A(T),1≤ji≤ni,1≤qp≤np}. If T is strongly connected, then Q is called a strong composition and if T is semicomplete, i.e., there is at least one arc between every pair of vertices, then Q is called a semicomplete composition.We obtain the following result: every strong digraph composition Q in which ni≥2 for every 1≤i≤t, has a good pair at every vertex of Q. The condition of ni≥2 in this result cannot be relaxed. We characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

A dominated pair condition for a digraph to be hamiltonian

In 1996, Bang-Jensen, Gutin, and Li proposed the following conjecture: If D is a strong digraph of order n where n≥2 with the property that d(x)+d(y)≥2n−1 for every pair of dominated non-adjacent vertices {x,y}, then D is hamiltonian. In this paper, we give an infinite family of counterexamples to this conjecture. In the same paper, they showed that for the above x,y, if they satisfy the condition either d(x)≥n, d(y)≥n−1 or d(x)≥n−1, d(y)≥n, then D is hamiltonian. It is natural to ask if there is an integer k≥1 such that every strong digraph of order n satisfying either d(x)≥n+k, d(y)≥n−1−k, or d(x)≥n−1−k, d(y)≥n+k, for every pair of dominated non-adjacent vertices {x,y}, is hamiltonian. In this paper, we show that k must be at most n−5 and prove that every strong digraph with k=n−4 satisfying the above condition is hamiltonian, except for one digraph on 5 vertices.

On the (di)graphs with (directed) proper connection number two

The (directed) proper connection number of a given (di)graph G is the least number of colors needed to edge-color G such that there exists a properly colored (di)path between every two vertices in G. There also exist vertex-coloring versions of the proper connection number in (di)graphs. We initiate the study of the complexity of computing the proper connection number and (two variants of) the proper vertex connection number, in undirected and directed graphs, respectively. First we disprove some conjectures of Magnant et al. (2016) on characterizing strong digraphs with directed proper connection number at most two. In particular, we prove that deciding whether a given digraph has directed proper connection number at most two is NP-complete. Furthermore, we show that there are infinitely many such digraphs without an even-length dicycle. To the best of our knowledge, the proper vertex connection number of digraphs has not been studied before. We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, on a more positive side we present polynomial-time recognition algorithms for bounded-treewidth graphs and bipartite graphs with proper connection number at most two.

Boundary Vertices of Cartesian Product of Directed Graphs

Suppose that $$D=(V,E)$$ is a strongly connected digraph. Let $$u, v \in V(D)$$ . The maximum distance md(u, v) defined as $$md(u,v)=max\{\overrightarrow{d}(u,v), \overrightarrow{d}(v,u)\}$$ is a metric, where $$\overrightarrow{d}(u,v)$$ denote the length of a shortest directed $$u-v$$ path in D. The boundary, contour, eccentricity and periphery sets of a strong digraph D are defined with respect to this metric. The main aim of this paper is to identify the above said metrically defined sets of a strong digraph D in terms of its prime factor decomposition with respect to Cartesian product.

Bipartite spanning sub(di)graphs induced by 2‐partitions

AbstractFor a given ‐partition of the vertices of a (di)graph , we study properties of the spanning bipartite subdigraph of induced by those arcs/edges that have one end in each . We determine, for all pairs of nonnegative integers , the complexity of deciding whether has a 2‐partition such that each vertex in (for ) has at least (out‐)neighbours in . We prove that it is ‐complete to decide whether a digraph has a 2‐partition such that each vertex in has an out‐neighbour in and each vertex in has an in‐neighbour in . The problem becomes polynomially solvable if we require to be strongly connected. We give a characterisation of the structure of ‐complete instances in terms of their strong component digraph. When we want higher in‐degree or out‐degree to/from the other set, the problem becomes ‐complete even for strong digraphs. A further result is that it is ‐complete to decide whether a given digraph has a ‐partition such that is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.

Structure of cycles in minimal strong digraphs

This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after suppressing the arcs of the cycle. By this process and by the contraction of all SCs into single vertices we obtain a Hasse diagram from the MSD. Among other properties, we show that any SC conformed by more than one vertex (non trivial SC) has at least one linear vertex (a vertex with indegree and outdegree equal to 1) in the MSD (Theorem 1); that in the Hasse diagram at least one linear vertex exists for each non trivial maximal (resp. minimal) vertex (Theorem 2); that if an SC contains a number λ of vertices of the cycle then it contains at least λ linear vertices in the MSD (Theorem 3); and, finally, that given a cycle of length q contained in the MSD, the number α of linear vertices contained in the MSD satisfies α≥⌊(q+1)∕2⌋ (Theorem 4).

Locally dense supereulerian digraphs

Let λ(D) be the arc strong-connectivity of a digraph D, and k>0 be an integer, and α,β be rational numbers. A strong digraph D is locally (α,β)+-arc-connected if ∀v∈V(D), λ(D[N+(v)])≥α|N+(v)|+β. A locally (0,k)+-arc-connected digraph is also called k+-locally-arc-connected. We show that for any integer k, a strong, k+-locally-arc-connected digraph may not be supereulerian, and we also show that every locally (23,0)+-arc-connected strong digraph is supereulerian.

Strong rainbow connection in digraphs

An arc-coloured digraph is strongly rainbow connected if for every pair of vertices (u,v) there exists a shortest path from u to v all of whose arcs have different colours. The strong rainbow connection number of a digraph is the minimum number of colours needed to make the graph strongly rainbow connected.In this paper, we study the strong rainbow connection number of minimally strongly connected digraphs, non-Hamiltonian strong digraphs and strong tournaments.

Structural properties of minimal strong digraphs versus trees

In this article, we focus on structural properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus (undirected) trees. For some of these properties, we give the matrix version, regarding nearly reducible matrices. We give bounds for the coefficients of the characteristic polynomial corresponding to the adjacency matrix of trees, and we conjecture bounds for MSDs. We also propose two different representations of an MSD in terms of trees (the union of a spanning tree and a directed forest; and a double directed tree whose vertices are given by the contraction of connected Hasse diagrams).

On the (di)graphs with (directed) proper connection number two

A coloring of a graph G is properly connected if every two vertices of G are the ends of a properly colored path. We study the complexity of computing the proper connection number (minimum number of colors in a properly connected coloring) for edge and vertex colorings, in undirected and directed graphs, respectively. First we disprove some conjectures of Magnant et al. (2016) on characterizing the strong digraphs with proper arc connection number at most two. Then, we prove that deciding whether a given digraph has proper arc connection number at most two is NP-complete. We initiate the study of proper vertex connectivity in digraphs and we prove similar results as for the arc version. Finally, we present polynomial-time recognition algorithms for bounded-treewidth graphs and bipartite graphs with proper edge connection number at most two.

3-Restricted arc connectivity of digraphs

The k-restricted arc connectivity of digraphs is a common generalization of the arc connectivity and the restricted arc connectivity. An arc subset S of a strong digraph D is a k-restricted arc cut if D−S has a strong component D′ with order at least k such that D−V(D′) contains a connected subdigraph with order at least k. The k-restricted arc connectivity λk(D) of a digraph D is the minimum cardinality over all k-restricted arc cuts of D.Let D be a strong digraph with order n≥6 and minimum degree δ(D). In this paper, we first show that λ3(D) exists if δ(D)≥3 and, furthermore, λ3(D)≤ξ3(D) if δ(D)≥4, where ξ3(D) is the minimum 3-degree of D. Next, we prove that λ3(D)=ξ3(D) if δ(D)≥n+32. Finally, we give examples showing that these results are best possible in some sense.

Disjoint Paths in Decomposable Digraphs

AbstractThe k‐linkage problem is as follows: given a digraph and a collection of k terminal pairs such that all these vertices are distinct; decide whether D has a collection of vertex disjoint paths such that is from to for . A digraph is k‐linked if it has a k‐linkage for every choice of 2k distinct vertices and every choice of k pairs as above. The k‐linkage problem is NP‐complete already for [11] and there exists no function such that every ‐strong digraph has a k‐linkage for every choice of 2k distinct vertices of D [17]. Recently, Chudnovsky et al. [9] gave a polynomial algorithm for the k‐linkage problem for any fixed k in (a generalization of) semicomplete multipartite digraphs. In this article, we use their result as well as the classical polynomial algorithm for the case of acyclic digraphs by Fortune et al. [11] to develop polynomial algorithms for the k‐linkage problem in locally semicomplete digraphs and several classes of decomposable digraphs, including quasi‐transitive digraphs and directed cographs. We also prove that the necessary condition of being ‐strong is also sufficient for round‐decomposable digraphs to be k‐linked, obtaining thus a best possible bound that improves a previous one of . Finally we settle a conjecture from [3] by proving that every 5‐strong locally semicomplete digraph is 2‐linked. This bound is also best possible (already for tournaments) [1].

Structural and spectral properties of minimal strong digraphs

In this article, we focus on structural and spectral properties of minimal strong digraphs (MSDs). We carry out a comparative study of properties of MSDs versus trees. This analysis includes two new properties. The first one gives bounds on the coefficients of characteristic polynomials of trees (double directed trees), and conjectures the generalization of these bounds to MSDs. As a particular case, we prove that the independent coefficient of the characteristic polynomial of a tree or an MSD must be −1, 0 or 1. For trees, this fact means that a tree has at most one perfect matching; for MSDs, it means that an MSD has at most one covering by disjoint cycles. The property states that every MSD can be decomposed in a rooted spanning tree and a forest of reversed rooted trees, as factors. In our opinion, the analogies described suppose a significative change in the traditional point of view about this class of digraphs.

The scrambling index set of primitive minimally strong digraphs

Denote the scrambling index set of primitive minimally strong digraphs of order n by SIn. For n≥13, we show that SIn contains all integers in the interval [3,K1(n)], whereK1(n)={n2+12n−138,if both n and n+12 are odd,n2+12n−58,if n is odd and n+12 is even,n2+10n−168,if n is even and n2 is odd,n2+10n−88,if both n and n2 are even. We also identify all elements of SIn which are larger than (n2−6n+13)/2 (n is odd) or (n2−6n+12)/2 (n is even). It is shown that there exist more “gaps” in the set SIn.

Dicycle Cover of Hamiltonian Oriented Graphs

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.

Characterization of Digraphic Sequences with Strongly Connected Realizations

AbstractLet be a sequence of of nonnegative integers pairs. If a digraph D with satisfies and for each i with , then d is called a degree sequence of D. If D is a strict digraph, then d is called a strict digraphic sequence. Let be the collection of digraphs with degree sequence d. We characterize strict digraphic sequences d for which there exists a strict strong digraph .