Testing a graph on 2-vertex-and 2-edge-connectivity are two fundamental algorithmic graph problems. For both problems, different lineartime algorithms with simple implementations are known. Here, an even simpler linear-time algorithm is presented that computes a structure from which both the 2-vertex-and 2-edge-connectivity of a graph can be easily "read off". The algorithm computes all bridges and cut vertices of the input graph in the same time.
Tutte proved that every 3-vertex-connected graph G on more than 4 vertices has a contractible edge. Barnette and Grünbaum proved the existence of a removable edge in the same setting. We show that the sequence of contractions and the sequence of removals from G to K 4 can be computed in O(|V | 2 ) time by extending Barnette's and Grünbaum's theorem. As an application, we derive a certificate for the 3-vertex-connectivity of graphs that can be easily computed and verified.
The USEPA 503 biosolids utilization rules recognize the need for higher than agronomic rate applications to mined lands under the assumption that N0 3-N contamination of groundwater will not be significant. We evaluated a range of biosolids loading rates (Ix to 7x agronomic rate of 14 Mg ha• 1) with and without added sawdust (to adjust the applied C:N ratio to approximately 20:1) on a reclaimed gravel mined soil and an undisturbed prime farmland soil for three growing seasons. The two experimental blocks were cropped to com (Zea mays) in 1996, and winter wheat (Triticum aestivum) and soybeans (Glycine max) in 1997. Root zone leachates were collected from zero-tension lysimeters under adjacent micro-plots. Effects of biosolids loading rate on crop yields were not as pronounced as expected due to relatively wet weather. Leachate NO,-N over the winter of96/97 increased incrementally (from< 20 to> 100 mg L• 1) with loading rate (Ix to 7x) and then declined sharply in March and April of 1997, finally approaching control level concentrations through the winter of 1997/1998 and beyond. Addition of sawdust significantly decreased NO,-N leachate levels at all biosolids loading rates except the 5x biosolids + sawdust treatment which exhibited a first winter spike in excess of 100 mg L• 1 • These data indicate that higher than agronomic loading rates ofbiosolids lead to enhanced N0 3-N leaching potentials over the first winter following application. However, this "one-time event" supports the original USEPA presumption that some net leaching under elevated loading rates is to be expected, but it is a short-term effect.
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