Maker-Maker and Maker-Breaker Games are PSPACE-Complete

We show that the problems of deciding the outcome of Maker-Maker and Maker-Breaker games played on arbitrary hypergraphs are PSPACE-complete. Maker-Breaker games have earlier been shown PSPACE-complete by Schaefer (1978); we give a simpler proof and show a reduction from Maker-Maker games to Maker-Breaker games

Chromatic Number in Time O(2.4023^n) Using Maximal Independent Sets

In this paper we improve an algorithm by Eppstein (2001) for finding the chromatic number of a graph. We modify the algorithm slightly, and by using a bound on the number of maximal independent sets of size  k from our recent paper (2003), we prove that the running time is O(2.4023^n). Eppstein's algorithm runs in time O(2.4150^n). The space usage for both algorithms is O(2^n)

On the Number of Maximal Bipartite Subgraphs of a Graph

We show new lower and upper bounds on the number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105^{n/10} ~= 1.5926^n such subgraphs, which improves an earlier lower bound by Schiermeyer (1996). We show an upper bound of n . 12^{n/4} ~= n . 1.8613^n and give an algorithm that lists all maximal induced bipartite subgraphs in time proportional to this bound. This is used in an algorithm for checking 4-colourability of a graph running within the same time bound

Maker-Maker and Maker-Breaker Games are PSPACE-Complete

We show that the problems of deciding the outcome of Maker-Maker and Maker-Breaker games played on arbitrary hypergraphs are PSPACE-complete. Maker-Breaker games have earlier been shown PSPACE-complete by Schaefer (1978); we give a simpler proof and show a reduction from Maker-Maker games to Maker-Breaker games

On the Number of Maximal Independent Sets in a Graph

We show that the number of maximal independent sets of size exactly k in any graph of size n is at most [ n/k ]^{k-(n mod k)} ([ n/k ] +1)^{n mod k}. For maximal independent sets of size at most k the same bound holds for k n/3 a bound of approximately 3^{n/3} is given. All the bounds are exactly tight and improve Eppstein (2001) who give the bound 3^{4k-n}4^{n-3k} on the number of maximal independent sets of size at most k, which is the same for n/

On the Number of Maximal Bipartite Subgraphs of a Graph

We show new lower and upper bounds on the number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105^{n/10} ~= 1.5926^n such subgraphs, which improves an earlier lower bound by Schiermeyer (1996). We show an upper bound of n . 12^{n/4} ~= n . 1.8613^n and give an algorithm that lists all maximal induced bipartite subgraphs in time proportional to this bound. This is used in an algorithm for checking 4-colourability of a graph running within the same time bound

New Algorithms for Exact Satisfiability

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O(2^{0.2325n}) and O(2^{0.1379n}), respectively. The previously best algorithms have running times O(2^{0.2441n}) for Exact Satisfiability (Monien, Speckenmeyer and Vornberger (1981)) and O(2^{0.1626n}) for Exact 3-Satisfiability (Kulikov and independently Porschen, Randerath and Speckenmeyer (2002)). We extend the case analyses of these papers and observe, that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time

Identification and management of interstitial lung abnormalities

Interstitial lung abnormalities (ILA) are incidentally observed specific CT findings in patients without clinical suspicion of interstitial lung disease (ILD). ILA with basal and peripheral predominance and features suggestive of fibrosis in more than 5% of any part of the lung should be referred for pulmonologist review. The strategy for monitoring as described in this review is based on clinical and radiological risk factors. ILA are associated with risk of progression to ILD and increased mortality. Early identification and assessment of risk factors for progression are essential to improve outcome.</p