Finite Digraph Research Articles

List-Compactness of Infinite Directed Graphs

A digraph H is homomorphically compact if the digraphs G which admit homomorphisms to H are exactly the digraphs whose finite subdigraphs all admit homomorphisms to H. In this paper we define a similar notion of compactness for list-homomorphisms. We begin by showing that it is essentially only finite digraphs that are compact with respect to list-homomorphisms. We then explore the effects of restricting the types of list-assignments which are permitted, and obtain some richer characterizations.

Construction of [formula omitted]-arc transitive digraphs

A digraph is k-arc transitive if it has a group of automorphisms which acts transitively on the set of k-arcs. Unlike the undirected case, in which the cycles are the only k-arc transitive finite graphs for k⩾8, there are k-arc transitive finite digraphs with arbitrary out-degree for every positive integer k. We show that every regular finite digraph admits a covering digraph which is k-arc transitive. The result provides a technique to construct k-arc transitive digraphs from arbitrary regular digraphs. Some examples are given from complete graphs with and without loops.

Tiling the Line with Triples

It is known the one dimensional prototile $0,a,a+b$ and its reflection $0,b,a+b$ always tile some interval. The subject has not received a great deal of further attention, although many interesting questions exist. All the information about tilings can be encoded in a finite digraph $D_{ab}$. We present several results about cycles and other structures in this graph. A number of conjectures and open problems are given.In [Go] an elegant proof by contradiction shows that a greedy algorithm will produce an interval tiling. We show that the process of converting to a direct proof leads to much stronger results.

Semikernels modulo F and kernels in digraphs

Let D be a finite digraph, V( D) and A( D) will denote the sets of vertices and arcs of D respectively. A kernel N of D is an independent set of vertices such that for every w∈ V( D)− N there exists an arc from w to N. Let F be a set of arcs of D (i.e. F⊆ A( D)), a set S⊆ V( D) is called a semikernel of D modulo F if S is an independent set of vertices such that for every z∈ V( D)− S for which there exists an Sz-arc of D− F, there also exists a zS-arc in D. In this paper is introduced the concept of semikernel modulo F and it is used to obtain a new sufficient condition for the existence of kernels in digraphs. As a consequence is obtained a generalization of the following result of B. Sands, N. Sauer and R. Woodrow; in case that the digraph is a finite digraph: Let D be a digraph whose arcs are coloured with two colors. If D contains no monochromatic infinite outward path, then there exists a set S of vertices of D such that: No two vertices of S are connected by a monochromatic directed path and for every vertex not in S there is a monochromatic directed path from x to a vertex in S.

Short proof of Menger's Theorem

A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a finite digraph for two subsets A and B of its vertex set is given.

An upper bound for the Hamiltonicity exponent of finite digraphs

Let D be a strongly k-connected digraph of order n ⩾ 2. We prove that for every l ⩾ n/2 k the power D 1 of D is Hamiltonian. Moreover, for any k ⩾ 1 and n > 2 k we exhibit a strongly k-connected digraph D of order n such that D ⌈ n 2k⌉−1 is non-Hamiltonian.

Lattices of processes for graphs

The structure of the lattice of processes of a finite digraph with entries is investigated. Necessary and sufficient conditions for the isomorphism of a given lattice to the lattice of processes of a certain digraph are obtained. Bibliography: 5 titles.

On topological tournaments of order 4 in digraphs of outdegree 3

It is proved that every finite digraph of minimum outdegree 3 contains a subdivision of the transitive tournament on 4 vertices. © 1996 John Wiley & Sons, Inc.

Exixtence of vertices of local connectivityk in digraphs of large outdegree

For every positive integerk, there is a positive integerf(k) such that every finite digraph of minimum outdegreef(k) contains verticesx, y joined byk openly disjoint paths.

An algorithm for optimum common root functions of two digraphs

Let G 1 and G 2 be finite digraphs, both with vertex set V. Suppose that each v of V has nonnegative integers f( v) and g( v) with f( v)⩽ g( v), and each arc e of G i has nonnegative integers a i ( e) and b i ( e) with a i ( e)⩽ b i ( e), i=1,2. In Cai (1990) a necessary and sufficient condition was given for the existence of k arborescences in G i covering each arc e of G i at least a i ( e) and at most b i ( e) times, i=1,2, and satisfying the condition that for each v in V f(v)⩽r 1(v)=r 2(v)⩽g(v) , where r i ( v) denotes the number of the arborescences in G i rooted at v. Such an r i is called a common root function and denoted by r. In this paper, we present a polynomial algorithm for finding an optimum common root function r for a given weight function defined on V.

Unordered Love in infinite directed graphs

A digraph D = (V, A) has the Unordered Love Property (ULP) if any two different vertices have a unique common outneighbor. If both (V, A) and (V, A−1) have the ULP, we say that D has the SDULP.A love‐master in D is a vertex ν0 connected both ways to every other vertex, such that D − ν0 is a disjoint union of directed cycles.The following results, more or less well‐known for finite digraphs, are proven here for D infinite: (i) if D is loopless and has the SDULP, then either D has a love‐master, or D is associable with a projective plane, obtainable by taking V as the set of points and the sets of outneighbors of vertices as the lines; (ii) every projective plane arises from a digraph with the SDULP, in this way.

Ecken von kleinem grad in kritisch n-fach zusammenhängenden digraphen

We prove that every critically n-connected, finite digraph has a vertex of outdegree less than 2 n or a vertex of indegree less than 2 n, but there is not always a vertex of outdegree less than 2 n. This bound 2 n−1 for the minimum of the out- and indegrees is, in general, best possible and was conjectured by Y. O. Hamidoune (“Contribution à l'étude de la connectivité d'un graphe”, Thèse d'Etat, Paris, 1980). An immediate consequence is the main result of the author's previous work ( Arch. Math. 25, 1974, 107–112) that every minimally n-edge-connected, finite digraph has a vertex of indegree and outdegree n.

Images of Rigid Digraphs

We characterize those finite digraphs which are homomorphic images of rigid graphs. In the infinite case we obtain partial results only and the problem seems to be difficult. This is related on the one side to the theory of well quasi-ordered sets and on the other side to universality of graphs of small height.

On critically connected digraphs

AbstractA digraph is called critically connected if it is connected, but the deletion of any vertex destroys the connectivity. We prove that every critically connected finite digraph has at least two vertices of outdegree one. As an application, we show that for n ≧ 2, there is no n‐connected, non‐complete, finite digraph such that the deletion of any n vertices results in a disconnected digraph.

Common root functions of two digraphs

AbstractLet D1 and D2 be finite digraphs, both with vertex set V, let a and b be given functions from V to Z+, and let k be a positive integer. In this paper we give a necessary and sufficient condition for the existence of k arc‐disjoint arborescences in each of D1 and D2 satisfying the condition that for each v in Vmagnified image a(v) <r1(v) = r2(v) < b(v).Where ri(v) denotes the number of the arborescences in Di rooted at v, i = 1, 2.

Highly Arc Transitive Digraphs

Finite digrahs Г with a group G of automorphisms acting transitively on the set of s-arcs, for some s ⩾ 2, are investigated. For each valency v and each s ⩾ 2 an infinite family of finite digraphs of valency v which are s-arc transitive but not (s + 1)-arc transitive are constructed. These examples are the result of a general construction given in the paper. Several classification results are proved and the case of G primitive on vertices is discussed.

A characterization of interval catch digraphs

It is shown that a (finite or infinite) digraph D is the catch digraph of a family of pointed intervals if and only if it contains no set of three vertices which fulfil: any two of them are weakly connected by a chain where no initial endpoint of an arc precedes the third vertex. Furthermore, it is shown that all these digraphs have weakly triangulated (and thus perfect) underlying graphs.

Sur les quasi-noyaux d'un graphe

We define three classes of quasi-kernels for a directed graph. As a consequence, we show the existence of quasi-kernels in every progressively finite graph and in every locally finite graph, generalizing the result of Chvátal and Lovász which deals with the finite case. Our method shows that the problem of finding a quasi-kernel in a finite digraph and the problem of finding the unique kernel of an acircuitic finite digraph have the same algorithmic complexity.

A counterexample to a generalization of Richardson's theorem

The following result has apparently been obtained in [2]: If a finite digraph has no circuits of length nk + r for n = 0, 1, 2,... and 0 < r < k, then it has a k-kernel ( k ⩾ 2; Richardson's theorem refers to the case k = 2). The present note shows that the result is not always valid for k > 2 unless an additional condition is imposed, such that the digraph be strongly connected [2].

Theory of annihilation games—I

Place tokens on distinct vertices of an arbitrary finite digraph with n vertices which may contain cycles or loops. Each of two players alternately selects a token and moves it from its present position u to a neighboring vertex v along a directed edge which may be a loop. If v is occupied, and u ≠ v, both tokens get annihilated and phase out of the game. The player first unable to move is the loser, the other the winner. If there is no last move, the outcome is declared a draw. An O( n 6) algorithm for computing the previous-player-winning, next-player-winning and draw positions of the game is given. Furthermore, an algorithm is given for computing a best strategy in O( n 6) steps and winning—starting from a next-player-winning position—in O( n 5) moves.