We give a definition of a bounded edge within the causal directed acyclic graph framework. A bounded edge generalizes the notion of a signed edge and is defined in terms of bounds on a ratio of survivor probabilities. We derive rules concerning the propagation of bounds. Bounds on causal effects in the presence of unmeasured confounding are also derived using bounds related to specific edges on...

Let G = (V,E) be any finite simple graph. A mapping C : E → [k] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced by all the edges which have either colour i or j is acyclic. The smallest number k of colours, such that G has an acy...

Counting the number of edge covers on graphs, denoted as the #Edge Covers problem, is well known to be #Pcomplete. In this paper, we present an algorithm that compute the number of edge covers in polynomial time if and only if the graph is acyclic. Our algorithm is based on a post-order traversal of the spanning tree of the original graph.

We provide a characterization of several graph parameters (the acyclic chromatic number, the arrangeability, and a sequence of parameters related to the expansion of a graph) in terms of forbidden subdivisions. Let us start with several definitions. Throughout the paper, we consider only simple undirected graphs. A graph G = sdt(G) is the t-subdivision of a graph G, if G is obtained from G by r...

Matrix M is k-concise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)-time algorithm to compute the row minima of any O(1)-concise n×m matrix. Our algorithm yields the first O(n+m)-time reductions from the replacement-paths problem on an n-node m-edge undirected graph (respectively, directed acyclic graph) to the single-source ...

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We describe an implementation of Italiano's partially dynamic data structure for maintaining transitive closure information in a directed graph and report on experimental results with random directed graphs and random sequences of operations. The operations supported are edge insertions or edge deletions, and queries. For the case of edge deletions the directed graph is assumed to be acyclic.