We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, k-trees and cops and robber games and study their natural translations to digraphs. We show that on digraphs the translations of these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kelly-width, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting potential applications including polynomial-time algorithms for NP-complete problems on graphs of bounded Kelly-width, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.
Abstract. Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures of such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width.
Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm design. Tree-width can be characterised by a graph searching game where a number of cops attempt to capture a robber. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure, called dag-width, that can be seen to describe how close a directed graph is to a directed acyclic graph (dag). We also provide an associated decomposition and show how it is useful for developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded dag-width. We also consider the relationship between dag-width and other connectivity measures such as directed tree-width and path-width. A consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomialtime computable on graphs of bounded dag-width.
Unhatched eggs and plasma samples from prefledged bald eagles (Haliaeetus leucocephalus) from a recovering population in the Canadian Great Lakes Basin were contaminated with organochlorine compounds at levels comparable to those reported for eagle populations in several areas of the United States. PCBs were among the most commonly detected contaminants and occurred at high concentrations in plasma. Plasma PCB concentrations in Lake Erie nestlings were significantly higher than in chicks from Lake Nipigon (0.130 and 0.047 mg/kg wet weight, respectively). Blood levels of mirex were higher in nestlings from Lake Superior compared to those from Lake Erie (0.0012 and 0.0006 mg/kg wet weight, respectively). Migration routes and over-winter locations of avian prey that constitute a part of the bald eagle chick's diet are likely to contribute to these spatial contaminant patterns in plasma. Atmospheric deposition and a cold condensation effect for chlordane compounds may have produced higher levels of these compounds in plasma samples from Lake Superior compared to samples from Lake Erie (0.020 and 0.008 mg/kg wet weight, respectively). Levels of DDE in plasma were generally low, ranging in concentration from 0.02 mg/kg wet weight for lakes Erie and Nipigon to 0.06 mg/kg wet weight for Lake Huron. Concentrations of organochlorines in eaglet plasma remained relatively stable between 1990 and 1996; no significant trends associated with year of sampling were detected. The data from Lake Erie showed no correlation between productivity and plasma levels of PCBs or DDE during this time period. There were no indications that the concentrations of contaminants detected were adversely affecting productivity in Canadian Great Lakes bald eagle populations. Residue levels in eggs from Lake Erie eagle territories were equally or more contaminated than eggs from other highly contaminated environments in the United States such as the Great Lakes and Columbia River estuary. Concentrations of DDE and PCBs in Lake Erie eggs declined significantly between 1974 and 1994. Although dieldrin and chlordane levels showed a similar decline, these relationships were not statistically significant. Mirex concentrations were relatively stable. Eggs from Lake of the Woods were significantly less contaminated than those from Lake Erie (PCBs: 3 and 84 mg/kg wet weight, respectively; DDE: 3 and 24 mg/kg wet weight, respectively). Reproductive success (number of young produced/active nest) remained constant between 1982 and 1996 and productivity is sufficient to maintain an increasing population. The number of active nests has increased during this period indicating that the population is expanding and might be expected to do so until suitable habitat becomes limiting. Reductions in organochlorine levels, reintroduction efforts, immigration from other populations, and changes in habitat quality have likely contributed to the observed growth in the number of breeding pairs.
Abstract. In this invited contribution [7], we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.
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