Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edgeindexed graphs’ (A′, i′) a...

We report on ASCA energy spectra of Cir X-1 taken near its zero phase on 1994 August 4–5. The ASCA SIS detectors allow a much more detailed study of the iron K complex than has been possible before. We find that prior to a sudden upward flux transition the dominant iron K feature appears to consist of a large edge from neutral or nearly-neutral iron. The depth of the edge corresponds to an abso...

In this work, we introduce the Cov-MECF framework, a special case of minimum-edge cost flow in which the input graph is bipartite. We observe that several important covering (and multi-covering) problems are captured in this unifying model and introduce a new heuristic LPO for any problem in this framework. The essence of LPO harnesses as an oracle the fractional solution in deciding how to gre...

We present approximation algorithms with O(n) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n) time algorithms of Ghosh. For simple polygon, there are O(n) visibility regions, thus any approximation algorithm for the set covering problem with approximation ratio of log(n) can be used for the approximation of n vertex and edge g...

The combinatorial Alexander dual of the independence complex Ind(G) and that of the edge covering complex EC(G) are shown to have isomorphic homology groups for each non-null graph G. This yields isomorphisms of homology groups of Ind(G) and EC(G) with homology dimensions being appropriately shifted and restricted. The results exhibits the complementary nature of homology groups of Ind(G) and E...

For r ≥ 2, α ≥ r − 1 and k ≥ 1, let c(r, α, k) be the smallest integer c such that the vertex set of any non-trivial r-uniform k-edge-colored hypergraph H with α(H) = α can be covered by c monochromatic connected components. Here α(H) is the maximum cardinality of a subset A of vertices in H such that A does not contain any edges. An old conjecture of Ryser is equivalent to c(2, α, k) = α(r − 1...

In this thesis, we investigate the problem of packing and covering odd (u, v)-trails in a graph. A (u, v)-trail is a (u, v)-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Given a graph G and two specified vertices u and v, the odd (u, v)-trail packing number, denoted by ν(u, v), is the maximum number of edge-...

In the thesis we study selected properties of random coverings of graphs introduced by Amit and Linial in 2002. A random n-covering of a graph G, denoted by G̃, is obtained by replacing each vertex v of G by an n-element set G̃v and then choosing, independently for every edge e = {x, y} ∈ E(G), uniformly at random a perfect matching between G̃x and G̃y. The first problem we consider is the typical ...