A cube decomposition Q of a graph G is said to be 2-perfect if for every edge {x, y} ∈ E(G), x and y are connected by a path of length 1 in exactly one cube of Q, and are also connected by a path of length 2 in exactly one (distinct) cube of Q. For both Kv and Kv −F , we give constructions for half of the cases which satisfy the obvious necessary conditions, with a small number of exceptions. W...

The vertex set of the resonance graph of a hexagonal graph G consists of 1-factors of G, two 1-factors being adjacent whenever their symmetric difference forms the edge set of a hexagon of G. A decomposition theorem for the resonance graphs of catacondensed hexagonal graph is proved. The theorem intrinsically uses the Cartesian product of graphs. A canonical binary coding of 1-factors of cataco...

We show how to decompose efficiently in parallel any graph into a number, γ̃, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. We achieve this decomposition in O(log n log log n) time using O(n + m) CREW PRAM processors, for an n-vertex, m-edge graph. This decomposition provides a general framework for solving graph problems efficiently in parallel. Its value i...

Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of nonnegative real weights to the copies of H in G such that for each e ∈ E(G), the sum of the weights of copies of H containing e is precisely one. An H-packing of a graph G is a set of edge disjoint copies of H in G. The following results are proved. For every fixed k > 2, every graph with n vertices and minim...

Let H be a tree on h ≥ 2 vertices. It is shown that if n is sufficiently large and G = (V,E) is an n-vertex graph with δ(G) ≥ bn/2c, then there are b|E|/(h − 1)c edge-disjoint subgraphs of G which are isomorphic to H. In particular, if h− 1 divides |E| then there is an H-decomposition of G. This result is best possible as there are infinitely many examples of trees on h vertices and graphs G wi...

Fourier ptychographic microscopy (FPM) is a promising computational imaging technique with high resolution, wide field-of-view (FOV) and quantitative phase recovery. So far, series of system errors that may corrupt the image quality FPM has been reported. However, an imperceptible artifact caused by edge effect caught our attention also degrade precision in cross-shape space. We found reconstru...

Peatlands provide multiple ecosystem services, including extensive carbon sequestration and storage, yet many peatlands have been degraded or destroyed. Peatlands' storage capacity is connected to inherently low decomposition rates, causing the buildup of organic matter. This pattern could be explained by waterlogged conditions that reduce amount available oxygen for decomposer community, a pH ...