On Kernels, β-graphs, and β-graph Sequences of Digraphs

We begin by investigating some conditions determining the existence of kernels in various classes of directed graphs, most notably in oriented trees, grid graphs, and oriented cycles. The question of uniqueness of these kernels is also handled. Attention is then shifted to $\gamma$-graphs, structures associated to the minimum dominating sets of undirected graphs. I define the $\beta$-graph of a given digraph analogously, involving the minimum absorbant sets. Finally, attention is given to iterative construction of $\beta$-graphs, with an attempt to characterize for what classes of digraphs these $\beta$-sequences terminate

Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs

Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more. We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion

Enumerating simple paths from connected induced subgraphs

We present an exact formula for the ordinary generating series of the simple paths, also called self-avoiding walks, between any two vertices of a graph. Our formula involves a sum over the connected induced subgraphs and remains valid on weighted and directed graphs. An intermediary result of our approach provides a unifying vision that conciliates several concepts used in the literature for counting simple paths. We obtain a relation linking the Hamiltonian paths and cycles of a graph to its connected dominating sets

Interval Routing Schemes for Circular-Arc Graphs

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we an outline algorithm to calculate the routing scheme for circular-arc graphs in O(n^2) time, where n is the number of vertices.Comment: 17 pages, to appear in "International Journal of Foundations of Computer Science