A kernel of a digraph is a set of vertices which is both independent and absorbant. Let D be a digraph such that every proper induced subdigraph has a kernel. If D has a kernel, then D is kernel perfect, otherwise D is critical kernel-imperfect (for short CKI-digraph). In this work we prove that if a CKI-digraph D is not 2-arc connected, then D − a is kernel perfect for any bridge a of D. If D ...

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. Besides, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows to relate properties of G and LG regarding the number of cycles of a given length.

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the c...

The eccentricity e(u) of a vertex u in a digraph G is the maximum distance from u to any other vertex in G. A vertex v is an eccentric vertex of u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G, but with an arc from u to v in ED(G) if and only if v is an eccentric vertex of u in G. Given a positive integer k, the k iterated ...

We prove a sharp Meyniel-type criterion for hamiltonicity of a balanced bipartite digraph: For a ≥ 2, a strongly connected balanced bipartite digraph D on 2a vertices is hamiltonian if d(u) + d(v) ≥ 3a whenever uv / ∈ A(D) and vu / ∈ A(D). As a consequence, we obtain a sharp sufficient condition for hamiltonicity in terms of the minimal degree: a strongly connected balanced bipartite digraph D ...

Graph labeling traces its origin to the famous conjecture that all trees are graceful presented by A.Rosa in 1966. In recent years, many research papers on graceful graph have been published. In this survey we have collected studies on the gracefulness of the directed circuit ~ Cm and the union graph of mutually non-intersecting ~ Cm, the digraph n · ~ Cm, the digraph n − ~ Cm, the union of the...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l ≤ 2c− 1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We s...

The Gallai-Milgram theorem says that the vertex set of any digraph with stability number k can be partitioned into k directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer k, there is a digraph D with stability number k such that deleting the vertices of any k− 1 directed paths in D leaves a digraph with ...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...