We study list homomorphism problems L-HOM(H) for the class of reflexive digraphs H (digraphs in which each vertex has a loop). These problems have been intensively studied in the case of undirected graphs H, and appear to be more difficult for digraphs. However, it is known that each problem L-HOM(H) is NP-complete or polynomial time solvable. In this paper we focus on reflexive digraphs. We in...

A seminal result by Lovász states that two digraphs A and B (possibly with loops) are isomorphic if and only if for every digraph X the number of homomorphisms X → A equals the number of homomorphisms X → B. Lovász used this result to deduce certain cancellation properties for the direct product of digraphs. We develop an analogous result for the class of digraphs without loops, and with weak h...

Interval digraphs were introduced by West et all. They can be recognized in polynomial time and admit a characterization in terms of incidence matrices. Nevertheless, we do not have a forbidden structure characterization nor a low-degree polynomial time algorithm. We introduce a new class of ‘adjusted interval digraphs’, obtained by a slight change in the definition. By contrast, these digraphs...

sponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their Hasse digraphs called KoDAGs. KoDAGs are special orderable Directed Acyclic Graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. KoDAGs appear to be distinguished family of Ferrers digraphs which are natural join of a corresponding ordering ch...

a kernel $j$ of a digraph $d$ is an independent set of vertices of $d$ such that for every vertex $w,in,v(d),setminus,j$ there exists an arc from $w$ to a vertex in $j.$in this paper, among other results, a characterization of $2$-regular circulant digraph having a kernel is obtained. this characterization is a partial solution to the following problem: characterize circulant digraphs which hav...

Let 1 ≤ n ∈ Z. Worst-case efficient dominating sets in digraphs are conceived so that their presence in certain strong digraphs ~ STn corresponds to that of efficient dominating sets in star graphs STn: The fact that the star graphs STn form a so-called dense segmental neighborly E-chain is reflected in a corresponding fact for the digraphs ~ STn. Related chains of graphs and open problems are ...

Cycle prefix digraphs are a class of Cayley coset graphs with many remarkable properties such as symmetry, large number of nodes for a given degree and diameter, simple shortest path routing, Hamiltonicity, optimal connectivity, and others. In this paper we show that the cycle prefix digraphs, like the Kautz digraphs, contain cycles of all lengths l, with l between two and N , the order of the ...

We describe a polynomial (O( n’.‘)) time algorithm DHAM for finding hamilton cycles in digraphs. For digraphs chosen uniformly at random from the set of digraphs with vertex set (1,2,. . . , n } and m = m(n) edges the limiting probability (as n + co) that DHAM finds a hamilton cycle equals the limiting probability that the digraph is hamiltonian. Some applications to random “ travelling salesma...

A simple decomposition for graphs yields generating functions for counting graphs by edges and connected components. A change of variables gives a new interpretation to the Tutte polynomial of the complete graph involving inversions of trees. The relation between the Tutte polynomial of the complete graph and the inversion enumerator for trees is generalized to the Tutte polynomial of an arbitr...