This thesis presents exact means for solving a family of NP-hard problems. Starting with the well-studied Exact Satisfiability problem (Xsat) parents, siblings and daughters are derived and studied, each with interesting practical and theoretical properties. While developing exact algorithms to solve the problems, we gain new insights into their structure and mutual similarities and differences...

in this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. we present sufficient conditions for the existence of long exact extended d- homology sequence. also we give some illustrative examples.

In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present sufficient conditions for the existence of long exact extended d- homology sequence. Also we give some illustrative examples.