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...

In this paper we initialize the study of independent domination in directed graphs. We show that an independent dominating set of an orientation of a graph is also an independent dominating set of the underlying graph, but that the converse is not true in general. We then prove existence and uniqueness theorems for several classes of digraphs including orientations of complete graphs, paths, tr...

In this paper we use the concept of integer-pair sequences, an invariant of graphs and digraphs introduced in Hakimi and Patrinos [9], and results on its unique realizability, in Das [4,5], to obtain results on the unique realizability of degree sequences, another invariant of graphs and digraphs. We thus present a unified approach to solving the problem of unique realizability of these two inv...

The multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dime...

The problem of distinguishing, in terms of graph topology, digraphs with real and partially non-real Laplacian spectra is important for applications. Motivated by the question posed in [R. Agaev, P. Chebotarev, Which digraphs with rings structure are essentially cyclic?, Adv. in Appl. Math. 45 (2010), 232–251], in this paper we completely list the Laplacian eigenvalues of some digraphs obtained...

We study the complexity of two dual covering and packing distance-based problems Broadcast Domination Multipacking in digraphs. A dominating broadcast a digraph D is function $$f:V(D)\rightarrow {\mathbb {N}}$$ such that for each vertex v D, there exists t with $$f(t)>0$$ having directed path to length at most f(t). The cost f sum f(v) over all vertices v. multipacking set S every integer d, ar...