A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges. The t-tessellability problem aims to decide whether there is a tessellation cover of the graph with t tessellations. This problem is motivated by its applications to quantum walk models, in especial, the evolution operator of the staggered model is obtained from a graph tessellation cover. We establish upper bounds on the tessellation cover number given by the minimum between the chromatic index of the graph and the chromatic number of its clique graph and we show graph classes for which these bounds are tight. We prove NP-completeness for t-tessellability if the instance is restricted to planar graphs, chordal (2, 1)-graphs, (1, 2)-graphs, diamond-free graphs with diameter five, or for any fixed t at least 3. On the other hand, we improve the complexity for 2-tessellability to a linear-time algorithm.

The present volume contains the contributions delivered at the 5th International Workshop on Fuzzy Logic and Applications (WILF 2003), hosted by the Istituto Italiano Studi Filosofici, Palazzo Serra di Cassano, Naples (Italy) and held on October 9-11, 2003.The volume includes the more recent achievements in the domain of theoretical, experimental and applied fuzzy logic and related techniques. To emphasize the particular connotation of the modern applications of fuzzy logic, special attention has been devoted to the recent trend of integrating and complementing fuzzy logic with rough set theory, neural networks, genetic algorithms and other formal theories and methodologies in order to define flexible and “intelligent” systems, based on the so-called paradigm of soft computing. The capabiblity of these techniques to incorporate imprecision and incomplete information, and to model complex systems, makes them …