A word-graph Gw is a digraph represented by a word w such that the vertex-set V(Gw) is the alphabet of w and the edge-set E(Gw) is determined by non-identical adjacent letter pairs in w. In this paper we study the strong-connectivity of word-graphs. Our main result is that the number of strongly connected word-graphs represented by l-words of over an n-alphabet can be expressed via a recurrence relation T(l,n) on the Stirling numbers of the second kind using a link between word partitions and digraph connectivity.
A digraph X=(V, E) is max-λ, if λ(X)=δ(X). A digraph X is super-λif every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all but a few exceptions, the strongly connected mixed Cayley digraphs are max-λand super-λ.
Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.
A vertex coloring of a strong digraph $D$ is a \emph{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 $smc_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $smc_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $smc_v(T)$.
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles …
We prove that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (we call such branchings 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 good no 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 e…
In this survey we overview known results on the strong subgraph $k$-connectivity and strong subgraph $k$-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and complexity, sharp bounds for strong subgraph $k$-(arc-)connectivity, minimally strong subgraph $(k, \ell)$-(arc-) connected digraphs. This survey contains several conjectures and open problems for further study.
Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc. The \emph{restricted arc-connectivity} $λ'(D)$ of a digraph $D$ is the minimum cardinality over all restricted arc-cuts of $D$. A strongly connected digraph $D$ is \emph{$λ'$-connected} when $λ'(D)$ exists. This paper presents a family $\cal{F}$ of strong digraphs of girth four that are not $λ'$-connected and for every strong digraph $D\notin \cal{F}$ with girth four it follows that it is $λ'$-connected.…
Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and an in-branching $B^-$ which are arc-disjoint (such two branchings are called a {\it good pair}), which settled a conjecture of Thomassen for digraphs of independence number 2. They also proved that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and gave an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. They asked for the smallest number $n$ of verti…
Let $p$ be a prime, $e$ a positive integer, $q = p^e$, and let $\mathbb{F}_q$ denote the finite field of $q$ elements. Let $f_i : \mathbb{F}_q^2\to\mathbb{F}_q$ be arbitrary functions, where $1\le i\le l$, $i$ and $l$ are integers. The digraph $D = D(q;\bf{f})$, where ${\bf f}=(f_1,\dotso,f_l) : \mathbb{F}_q^2\to\mathbb{F}_q^l$, is defined as follows. The vertex set of $D$ is $\mathbb{F}_q^{l+1}$. There is an arc from a vertex ${\bf x} = (x_1,\dotso,x_{l+1})$ to a vertex ${\bf y} = (y_1,\dotso,y_{l+1})$ if $ x_i + y_i = f_{i-1}(x_1,y_1) $ for all $i$, $2\le i \le l+1$. In this paper we study t…
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
A graph $G=(V,E)$ is representable if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. If $W$ is $k$-uniform (each letter of $W$ occurs exactly $k$ times in it) then $G$ is called $k$-representable. It was shown that a graph is representable if and only if it is $k$-representable for some $k$. Minimum $k$ for which a representable graph $G$ is $k$-representable is called its representation number. In this paper we give a characterization of representable graphs in terms of orientations. Namely, we show …
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
Figueroa-Centeno et al. introduced the following product of digraphs: let $D$ be a digraph and let $Γ$ be a family of digraphs such that $V(F)=V$ for every $F\in Γ$. Consider any function $h:E(D)\longrightarrowΓ$. Then the product $D\otimes_{h} Γ$ is the digraph with vertex set $V(D)\times V$ and $((a,x),(b,y))\in E(D\otimes_hΓ)$ if and only if $(a,b)\in E(D)$ and $(x,y)\in E(h (a,b))$. In this paper, we introduce the undirected version of the $\otimes_h$-product, which is a generalization of the classical direct product of graphs and, motivated by it, we also recover a generalization of the c…
Eccentric Connectivity Index of Cartesian product and Strong product of Strongly Connected Digraphs
NSS College Ottappalam（N.S.S. 奥特帕拉姆学院）；Department of Mathematics, Government College Chittur（奇图尔政府学院数学系）；University of Calicut（卡利卡特大学）；Department of Collegiate Education, Government of Kerala（喀拉拉邦 collegiate 教育部）
Two previous papers, arXiv:1803.00284 and arXiv:1803.00281, introduced and studied strong subgraph $k$-connectivity of digraphs obtaining characterizations, lower and upper bounds and computational complexity results for the new digraph parameter. The parameter is an analog of well-studied generalized $k$-connectivity of undirected graphs. In this paper, we introduce the concept of strong subgraph $k$-arc-connectivity of digraphs, which is an analog of generalized $k$-edge-connectivity of undirected graphs. We also obtain characterizations, lower and upper bounds and computational complexity r…
Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $α\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ The largest modulus of the eigenvalues of $A_α(G)$ is called the $A_α$ spectral radius of $G$. In this paper, we determine the digraphs which attain the maximum (or minimum) $A_α$ spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also discuss a number of ope…
Let $M$ be a representable matroid, and $Q, R, S, T$ subsets of the ground set. We prove that, if $M$ is sufficiently large, then there is an element $e$ such that deleting or contracting $e$ preserves both the $Q$-$R$ and the $S$-$T$ connectivities. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved.
By generalizing a recent result of Hong, Liu, and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple $k$-node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple.
Generalizing well-known results of Erdős and Lovász, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $λ(H)\geq \lceil{}\frac{k-1}{k}λ(G)\rceil$, where $λ(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs 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 infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that…
Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. A strong subgraph $H$ of $D$ is called an $S$-strong subgraph if $S\subseteq V(H)$. A pair of $S$-strong subgraphs $D_1$ and $D_2$ are said to be arc-disjoint if $A(D_1)\cap A(D_2)=\emptyset$. Let $λ_S(D)$ be the maximum number of arc-disjoint $S$-strong subgraphs in $D$. The strong subgraph $k$-arc-connectivity is defined as $$λ_k(D)=\min\{λ_S(D)\mid S\subseteq V(D), |S|=k\}.$$ The parameter $λ_k(D)$ can be seen as a generalization of classical edge-connectivity of undirected graphs. In this paper, we…
In this paper, we show that the edge connectivity of a distance-regular digraph $Γ$ with valency $k$ is $k$ and for $k>2$, any minimum edge cut of $Γ$ is the set of all edges going into (or coming out of) a single vertex. Moreover we show that the same result holds for strongly regular digraphs. These results extend the same known results for undirected case with quite different proofs.
Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivit…
A rooted digraph is a vertex-flame if for every vertex $v$ there is a set of internally disjoint directed paths from the root to $v$ whose set of terminal edges covers all ingoing edges of $v$. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by the third author and the analogue of Lovász' result for countable digraphs was shown. We strengthen…
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, a directed $(S, r)$-Steiner path or, simply, an $(S, r)$-path is a directed path $P$ started at $r$ with $S\subseteq V(P)$. Two $(S, r)$-paths are said to be arc-disjoint if they have no common arc. Two arc-disjoint $(S, r)$-paths are said to be internally disjoint if the set of common vertices of them is exactly $S$. Let $κ^p_{S,r}(D)$ (resp. $λ^p_{S,r}(D)$) be the maximum number of internally disjoint (resp. arc-disjoint) $(S, r)$-paths in $D$. The directed path $k$-connectivity of $D$ is defined as $$…
Let $D=(V,A)$ be a digraph of order $n$, $S$ a subset of $V$ of size $k$ and $2\le k\leq n$. Strong subgraphs $D_1, \dots , D_p$ containing $S$ are said to be internally disjoint if $V(D_i)\cap V(D_j)=S$ and $A(D_i)\cap A(D_j)=\emptyset$ for all $1\le i<j\le p$. Let $κ_S(D)$ be the maximum number of internally disjoint strong digraphs containing $S$ in $D$. The strong subgraph $k$-connectivity is defined as $$κ_k(D)=\min\{κ_S(D)\mid S\subseteq V, |S|=k\}.$$ A digraph $D=(V, A)$ is called minimally strong subgraph $(k,\ell)$-connected if $κ_k(D)\geq \ell$ but for any arc $e\in A$, $κ_k(D-e)\leq…
By a well known theorem of Robbins, a graph $G$ has a strongly connected orientation if and only if $G$ is 2-edge-connected and it is easy to find, in linear time, either a cut edge of $G$ or a strong orientation of $G$. A result of Durand de Gevigny shows that for every $k\geq 3$ it is NP-hard to decide if a given graph $G$ has a $k$-strong orientation. Thomassen showed that one can check in polynomial time whether a given graph has a 2-strong orientation. This implies that for a given digraph $D$ we can determine in polynomial time whether we can reorient (=reverse) some arcs of $D=(V,A)$ to…
In this paper, we introduce a corresponding between bipartite graphs with a perfect matching and digraphs, which implicates an equivalent relation between the extendibility of bipartite graphs and the strongly connectivity of digraphs. Such an equivalent relation explains the similar results on $k$-extendable bipartite graphs and $k$-strong digraphs. We also study the relation among $k$-extendable bipartite graphs, $k$-strong digraphs and combinatorial matrices. For bipartite graphs that are not 1-extendable and digraphs that are not strong, we prove that the elementary components and strong c…
Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connecti…
An arc-coloured digraph $D$ is said to be \emph{rainbow connected} if for every two vertices $u$ and $v$ there is an $uv$-path all whose arcs have different colours. The minimun number of colours required to make the digraph rainbow connected is called the \emph{rainbow connection number} of $D$, denoted $\stackrel{\rightarrow}{rc}(D)$. In \cite{Dorbec} it was showed that if $T$ is a strong tournament with $n\geq 5$ vertices, then $2\leq \stackrel{\rightarrow}{rc}(T)\leq n-1$; and that for every $n$ and $k$ such that $3\leq k\leq n-1$, there exists a tournament $T$ on $n$ vertices such that $\…
A particular case of Caccetta-Häggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any …
Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1, \dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set $\{u_{i,j_i}\colon\, 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\colon\, u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc between every pair of vertices. Digraph compositions generalize s…
This paper is the first from a series of papers that establish a common analogue of the strong component and basilica decompositions for bidirected graphs. A bidirected graph is a graph in which a sign $+$ or $-$ is assigned to each end of each edge, and therefore is a common generalization of digraphs and signed graphs. Unlike digraphs, the reachabilities between vertices by directed trails and paths are not equal in general bidirected graphs. In this paper, we set up an analogue of the strong connectivity theory for bidirected graphs regarding directed trails, motivated by factor theory. We …
If $G$ is a group acting on a set $Ω$ and $α, β\in Ω$, the digraph whose vertex set is $Ω$ and whose arc set is the orbit $(α, β)^G$ is called an {\em orbital digraph} of $G$. Each orbit of the stabiliser $G_α$ acting on $Ω$ is called a {\it suborbit} of $G$. A digraph is {\em locally finite} if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph $Γ$ has more than one end if there exists a finite set of vertices $X$ such that the induced digraph $Γ\setminus X$ contains at least two infinite connected components; if there exists such a set containing precis…
Universidade de São Paulo（圣保罗大学）；Universidade Federal do ABC（ABC联邦大学）；Universidad de Chile（智利大学）；University of Chile（智利大学）；Universidad Andres Bello（安德烈斯贝略大学）
We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc {\it deletable} if its deletion leaves a strongly connected digraph. Given a $3$-edge-connected graph $G$, we define its Frank number $f(G)$ to be the minimum number $k$ such that there exist $k$ orientations of $G$ with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that $f(G)…
A deterministic finite automaton (DFA) of $n$ states over a $k$-letter alphabet can be seen as a digraph with $n$ vertices which all have exactly $k$ labeled out-arcs ($k$-out digraph). In 1973 Grusho first proved that with high probability (whp) in a random $k$-out digraph there is a strongly connected component (SCC) of linear size that is reachable from all vertices, i.e., a giant. He also proved that the size of the giant follows a central limit law. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of overlapping tree-like structures. Thus…
We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give a Sabidussi-type characterization in the case of monoids. We then correct a proof of Zelinka from '81 that characterizes semigroup digraphs with outdegree $1$. Further, answering a question of Knauer and Knauer, we construct for every $k\geq 2$ connected $k$-outregular non-semigroup digraphs. On the other hand, we show that every sink-free directed graph is a union of connected components of a monoid digraph. Second, we consider monoid graphs…
