We present a graph-theoretic model of discourse based on the Questions Under Discussion (QUD) framework. Questions and assertions are treated as edges connecting discourse states in a rooted graph, modeling the introduction and resolution of various QUDs as paths through this graph. The amount of common ground presupposed by interlocutors at any given point in a discourse corresponds to graphic...

Rooted quivers are quivers that do not contain A∞ ≡ ··· → • → • as a subquiver. The existence of flat covers and cotorsion envelopes for representations of these quivers have been studied by Enochs et al. The main goal of this paper is to prove that flat covers and cotorsion envelopes exist for representations of A∞. We first characterize finitely generated projective representations of A∞. We ...

7 An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is an ‘antipodal’ concept of matroid. 9 We shall show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S7 where S7 is the smallest semimodular lattice that is not 11 modular. It is also shown ...

When G is a rooted graph where each edge may independently succeed with probability p, we consider the expected number of vertices in the operational component of G containing the root. This expected value EV (G;p) is a polynomial in p. We present several distinct equivalent formulations of EV (G;p), unifying prior treatments of this topic. We use results on network resilience (introduced by Co...

For a rooted graph G, let EVb(G; p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EVb(G; p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of E...

For a rooted graph G, let EV (G; p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We examine combinatorial properties of this polynomial, proving that G is k-edge connected iff EV ′(G; 1) = · · · = EV k−1(G; 1) = 0. We find bounds on the first and second derivatives of EV (G; p); applications yield character...

For finite q, we classify the countable, descendant-homogeneous digraphs in which the descendant set of any vertex is a q-valent tree. We also give conditions on a rooted digraph Γ which allow us to construct a countable descendant-homogeneous digraph in which the descendant set of any vertex is isomorphic to Γ. 2010 Mathematics Subject Classification: 05C20, 05C38, 20B27

The expected number of vertices that remain joined to the root vertex s of a rooted graph Gs when edges are prone to fail is a polynomial EV (Gs; p) in the edge probability p that depends on the location of s. We show that optimal locations for the root can vary arbitrarily as p varies from 0 to 1 by constructing a graph in which every permutation of k specified vertices is the ‘optimal’ orderi...