It is shown that in the random graph Gn,p with (fixed) edge probability p > 0, the number of edges that have to be examined in order to identify an acyclic orientation is Θ(n log n) almost surely. For unrestricted p, an upper bound of O(n log n) is established. Graphs G = (V,E) in which all edges have to be examined are considered, as well.

We present near-optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. Our technique improves the existing bounds for these problems on directed acyclic graphs, planar graphs, and non-planar integer-edge-weighted ...

The problem of distances in a graph, G, is one of the most studied questions, both from theoretical point of view and applications (the reader can consult two recent reviews ). It is connected to the Wiener number, W, or "the path number" as denominated by its initiator. In acyclic structures, Wiener number and its extension, hyper-Wiener number, can be defined as edge/path contributions, Ie/p ...

Abstract. Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientatio...

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n2.75) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m 3 2 log n,m 32 + n log n}) by Katriel and Bodlaender. We also provide an empirical comparison of our al...

If D = (V,A) is a digraph, its niche hypergraph NH(D) = (V, E) has the edge set E = {e ⊆ V | |e| ≥ 2 ∧ ∃v ∈ V : e = N− D (v) ∨ e = N + D (v)}. Niche hypergraphs generalize the well-known niche graphs (cf. [?]) and are closely related to competition hypergraphs (cf. [?]) as well as double competition hypergraphs (cf. [?]). We present several properties of niche hypergraphs of acyclic digraphs.

We show the existence of various versions of expander graphs using Kolmogorov complexity. This method seems superior to the usual “probabilistic construction”. Also, the best known bounds on the size of expanders and superconcentrators can be obtained this way. In the case of (acyclic) superconcentrators we obtain the density 34. Also, we review related graph properties, like magnification, edg...

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n) time, independent of the number m of edges inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m 3 2 logn, m 3 2 + n logn}) by Katriel and Bodlaender. We also provide an empirical comparison of our algori...

The vertex [edge] arboricity a(G) [a1(G)] of a graph G is the minimum number of subsets into which V (G) [E(G)] can be partitioned so that each subset induces an acyclic subgraph. Let G(m,n) be the class of connected simple graphs of order n and size m and let π ∈ {a, a1}. In this paper we determine π(m,n) := {π(G) : G ∈ G(m,n)} for integers m,n such that n− 1 ≤ m ≤ (n 2 ) .

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in O(n) time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of O(min{m 2 log n,m 3 2 + n log n}) by Katriel and Bodlaender. We also provide an empirical comparison of our algorit...