Transitive Digraphs Research Articles

Unions of digraphs which become kernel perfect

A kernel of a digraph, D, is a set of vertices, K, which is both independent: every two vertices are not adjacent; and absorbent: for every vertex in V(D)∖K there exists an ex-neighbor in K. A digraph is said to be kernel-perfect if and only if any induced subdigraph has a kernel. A classical result obtained by Sands, Sauer and Woodrow in 1982 asserts that the union of two transitive digraphs is a kernel-perfect digraph. A digraph D is transitive whenever the existence of the arcs (u,v) and (v,w) in D implies the existence of the arc (u,w) in D. Later, it was studied the union of a right-pretransitive digraph and a left-pretransitive digraph, and how it becomes a kernel-perfect digraph (Galeana-Sánchez and Rojas-Monroy, 2004). In this paper we obtain sufficient conditions for the union of two kernel-perfect digraphs becomes a kernel perfect digraph. And these conditions are tight. As a consequence many previous results are generalized.

Descendant sets in infinite, primitive highly arc transitive digraphs with prime power out-valency

The descendant set desc(α) of a vertex α in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from α. We study the structure of descendant sets Γ in an infinite, primitive, highly arc transitive digraph with out-valency pk, where p is a prime and k≥1. It was already known that Γ is a tree when k=1 and we show the same holds when k=2. However, for k≥3 there are examples of infinite, primitive highly arc transitive digraphs of out-valency pk whose descendant sets are not trees, for some prime p.

On The Enumeration of The Transitive and Acyclic Digraphs Having a Fixed Support Set

In previous works of the author, the concept of a binary reflexive adjacency relations was introduced on the set of all binary relations of the set , and an algebraic system consisting of all binary relations of the set and of all unordered pairs of adjacent binary relations was defined. If is a finite set, then this algebraic system is a graph (graph of binary relations ). The current paper introduces the notion of a support set for acyclic and transitive digraphs. This is the collections and consisting of the vertices of the digraph that have zero indegree and zero outdegree, respectively. It is proved that if is a connected component of the graph containing the acyclic or transitive digraph , then . A formula for the number of acyclic and transitive digraphs having a fixed support set is obtained.

Connectivity of half vertex transitive digraphs

A bipartite digraph is said to be a half vertex transitive digraph if its automorphism acts transitively on the sets of its bipartition, respectively. In this paper, bipartite double coset digraphs of groups are defined and it is shown that any half vertex transitive digraph is isomorphic to some half double coset digraph, and we show that the connectivity of any strongly connected half transitive digraph is its minimum degree.

Об опорных множествах ациклических и транзитивных ориентированных графов

Об опорных множествах ациклических и транзитивных ориентированных графов

Infinite quasi-transitive digraphs with domination number [formula omitted

In 2014 D. Pálvölgyi and A. Gyárfás explored the minimum dominating set of a digraph with an arc partition into transitive digraphs. A. Gyárfás proposed the conjecture “for each positive integerk, there exists a (least)p(k)such that everyk-transitive tournament has a dominating set of at mostp(k)vertices” a special case of a conjecture of Erdős, Sands, Sauer and Woodrow, and they show its relations with other conjectures as well as important consequences in case the conjecture is true. We consider the vertex version of this problem and prove that a finite or infinite quasi-transitive digraph with no infinite inward paths and a vertex partition into k-transitive tournaments has a dominating set of order at most two, if the vertex partition has at most two infinite classes, no heterochromatic directed triangles and δ−(D′)≥1, for every induced digraph D′ on three classes of the vertex partition. This result also holds for tournaments, which are quasi-transitive digraphs. Finally, we show that the conditions are tight.

Colourings, homomorphisms, and partitions of transitive digraphs

We investigate the complexity of generalizations of colourings (acyclic colourings, (k,ℓ)-colourings, homomorphisms, and matrix partitions), for inputs restricted to the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.

Universal and Near-Universal Cycles of Set Partitions

We study universal cycles of the set $\mathcal{P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with $\mathcal{P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or ucycles) exist, since vertices represent equivalence classes of partitions. We use this result to prove, however, that ucycles of $\mathcal{P}(n,k)$ exist for all $n \geq 3$ when $k=2$. We reprove that they exist for odd $n$ when $k = n-1$ and that they do not exist for even $n$ when $k = n-1$. An infinite family of $(n,k)$ for which ucycles do not exist is shown to be those pairs for which ${{n-2}\brace{k-2}}$ is odd ($3 \leq k < n-1$). We also show that there exist universal cycles of partitions of $[n]$ into $k$ subsets of distinct sizes when $k$ is sufficiently smaller than $n$, and therefore that there exist universal packings of the partitions in $\mathcal{P}(n,k)$. An analogous result for coverings completes the investigation.

Model Checking Existential Logic on Partially Ordered Sets

We study the problem of checking whether an existential sentence (i.e., a first-order sentence in prefix form built using existential quantifiers and all Boolean connectives) is true in a finite partially ordered set (a poset). A poset is a reflexive, antisymmetric, and transitive digraph. The problem encompasses the fundamental embedding problem of finding an isomorphic copy of a poset as an induced substructure of another poset. Model checking existential logic is already NP-hard on a fixed poset; thus, we investigate structural properties of posets yielding conditions for fixed-parameter tractability when the problem is parameterized by the sentence. We identify width as a central structural property (the width of a poset is the maximum size of a subset of pairwise incomparable elements); our main algorithmic result is that model checking existential logic on classes of finite posets of bounded width is fixed-parameter tractable. We observe a similar phenomenon in classical complexity, in which we prove that the isomorphism problem is polynomial-time tractable on classes of posets of bounded width; this settles an open problem in order theory. We surround our main algorithmic result with complexity results on less restricted, natural neighboring classes of finite posets, establishing its tightness in this sense. We also relate our work with (and demonstrate its independence of) fundamental fixed-parameter tractability results for model checking on digraphs of bounded degree and bounded clique-width.

Infinite primitive and distance transitive directed graphs of finite out-valency

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.

-TRANSITIVE DIGRAPHS PRESERVING A CARTESIAN DECOMPOSITION

In this paper, we combine group-theoretic and combinatorial techniques to study$\wedge$-transitive digraphs admitting a cartesian decomposition of their vertex set. In particular, our approach uncovers a new family of digraphs that may be of considerable interest.

On a cohomology of digraphs and Hochschild cohomology

We use a differential form cohomology theory on transitive digraphs to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

Highly arc-transitive digraphs — Structure and counterexamples

Two problems of Cameron, Praeger, and Wormald [Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica (1993)] are resolved. First, locally finite highly arc-transitive digraphs with universal reachability relation are presented. Second, constructions of two-ended highly arc-transitive digraphs are provided, where each `building block' is a finite bipartite digraph that is not a disjoint union of complete bipartite digraphs. Both of these were conjectured impossible in the above-mentioned paper. We also describe the structure of two-ended highly arc-transitive digraphs in more generality, heading towards a characterization of such digraphs. However, the complete characterization remains elusive.

Fully simple semihypergroups, transitive digraphs, and sequence A000712

Fully simple semihypergroups have been introduced in [9], motivated by the study of the transitivity of the fundamental relation β in semihypergroups. Here, we determine a transversal of isomorphism classes of fully simple semihypergroups with a right absorbing element. The structure of that transversal can be described by means of certain transitive, acyclic digraphs. Moreover, we prove that, if n is an integer ≥2, then the number of isomorphism classes of fully simple semihypergroups of size n+1, with a right absorbing element, is the (n+1)-th term of sequence A000712 in [20], namely, ∑k=0np(k)p(n−k), where p(k) denotes the number of nonincreasing partitions of integer k.

Reachability relations, transitive digraphs and groups

In A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566–1581, it was shown that properties of digraphs such as growth, property Z , and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if a transitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth.

H-Paths and H-Cycles in H-Coloured Digraphs

A digraph D is a strongly transitive digraph if for any vertices $${u, v, w \in V(D)}$$ u , v , w ? V ( D ) (possibly u = w) such that $${\{(u, v), (v, w)\} \subseteq A(D)}$$ { ( u , v ) , ( v , w ) } ⊆ A ( D ) implies $${(u, w) \in A(D)}$$ ( u , w ) ? A ( D ) (when u = w and $${\{(u, v), (v, w)\} \subseteq A(D)}$$ { ( u , v ) , ( v , w ) } ⊆ A ( D ) then $${\{(u, u), (v, v)\} \subseteq A(D)}$$ { ( u , u ) , ( v , v ) } ⊆ A ( D ) ). Let H be a strongly transitive digraph (possibly with loops) and D a digraph(that contains neither loops nor multiple arcs). A digraph D is said to be H-coloured if the arcs of D are coloured with the vertices of H. Will be denoted by c(x, y) the color of the arc $${(x, y)\in A(D)}$$ ( x , y ) ? A ( D ) . A directed walk(directed path) $${C=(z_{0}, z_{1}, \ldots, z_{t})}$$ C = ( z 0 , z 1 , ? , z t ) in D will be called an H-walk(path) in D if $${(c(z_{0}, z_{1}), c(z_{1}, z_{2}), \ldots, c(z_{t-1}, z_{t}))}$$ ( c ( z 0 , z 1 ) , c ( z 1 , z 2 ) , ? , c ( z t - 1 , z t ) ) is a directed walk in H. Let D 1 and D 2 be spanning subdigraphs of D. A succession [u, v, w, u] is a (D 1, D, D 2) H-subdivision of C 3, if there exist: T 1 an H-path from u to v contained in D 1, T an H-path from v to w in D, T 2 an H-path from w to u contained in D 2; and these path satisfies: (1) (c(final arc of T 1), c(initial arc of T)) $${\notin A(H)}$$ ? A ( H ) , (c(final arc of T), c(initial arc of T 2)) $${\notin A(H)}$$ ? A ( H ) and (c(final arc of T 2), c(initial arc of T 1)) $${\notin A(H)}$$ ? A ( H ) ; (2) $${T_{1} \bigcup T\bigcup T_{2}}$$ T 1 ? T ? T 2 is a cycle in D. A succession [u, v, w, x ] is a (D 1, D, D 2) H-subdivision of P 3, if there exist: T 1 an H-path from u to v contained in D 1, T an H-path from v to w in D, T 2 an H-path from w to x contained in D 2; such that (1) (c(final arc of T 1), c(initial arc of T)) $${\notin A(H)}$$ ? A ( H ) and (c(final arc of T), c(initial arc of T 2)) $${\notin A(H)}$$ ? A ( H ) ; (2) $${T_{1} \bigcup T\bigcup T_{2}}$$ T 1 ? T ? T 2 is a path in D. Let H be a strongly transitive digraph and D an H-coloured digraph. Let D 1 and D 2 be spanning subdigraphs of D. Will be said that P = {D 1, D 2} is an H-separation of D if: (1) $${A(D_{1}) \bigcap A(D_{2})= \emptyset, A(D_{1}) \bigcup A(D_{2}) = A(D)}$$ A ( D 1 ) ? A ( D 2 ) = ? , A ( D 1 ) ? A ( D 2 ) = A ( D ) ; (2) every H-path of D is contained in D 1 or it is contained in D 2. In this paper will be proved that: if H is a strongly transitive digraph and D is an H-coloured digraph, P = {D 1, D 2} an H-separation of D such that: (1) every cycle of D that is contained in D i is an H-cycle for $${i\in \{1, 2\}}$$ i ? { 1 , 2 } ; (2) D does not contain a (D 1, D, D 2) H-subdivision of C 3; (3) if (u, z, w, x 0) is a (D 1, D, D 2) H-subdivision of P 3 then there exists some H-path between u and x 0. Then D has an H-kernel by paths.

On the structure of strong k-transitive digraphs

A digraph D is k-transitive if the existence of a directed path (v0, v1, . . . , vk), of length k implies that (v0, vk) ∈ A(D). Clearly, a 2-transitive digraph is a transitive digraph in the usual sense. Transitive digraphs have been characterized as compositions of complete digraphs on an acyclic transitive digraph. Also, strong 3 and 4-transitive digraphs have been characterized. In this work we analyze the structure of strong k-transitive digraphs having a cycle of length at least k. We show that in most cases, such digraphs are complete digraphs or cycle extensions. Also, the obtained results are used to prove some particular cases of the Laborde-Payan-Xuong Conjecture.

On the existence of (k,l)-kernels in infinite digraphs: A survey

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N , u 6= v, then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k− 1)-kernel. This work is a survey of results proving sufficient conditions for the existence of (k, l)-kernels in infinite digraphs. Despite all the previous work in this direction was done for (2, 1)-kernels, we present many original results concerning (k, l)-kernels for distinct values of k and l. The original results are sufficient conditions for the existence of (k, l)kernels in diverse families of infinite digraphs. Among the families that we study are: transitive digraphs, quasi-transitive digraphs, right/left pretransitive digraphs, cyclically k-partite digraphs, κ-strong digraphs, k-transitive digraphs, k-quasi-transitive digraphs.

Constructing highly arc transitive digraphs using a direct fibre product

We introduce a construction of highly arc transitive digraphs using a direct fibre product. This product generalizes some known classes of highly arc transitive digraphs but also allows us to construct new ones. We use the product to obtain counterexamples to a conjecture advanced by Cameron, Praeger and Wormald on the structure of certain highly arc transitive digraphs.

4-transitive digraphs I: the structure of strong 4-transitive digraphs

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v, w ∈ V (D), (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈ A(D). A very useful structural characterization of transitive digraphs has been known for a long time, and recently, 3-transitive digraphs have been characterized. In this work, some general structural results are proved for k-transitive digraphs with arbitrary k ≥ 2. Some of this results are used to characterize the family of 4-transitive digraphs. Also some of the general results remain valid for k-quasi-transitive digraphs considering an additional hypothesis. A conjecture on a structural property of k-transitive digraphs is proposed.