  A phylogenetic tree is a graphical representation of an evolutionary history
of taxa in which the leaves correspond to the taxa and the non-leaves
correspond to speciations. One of important problems in phylogenetic analysis
is to assemble a global phylogenetic tree from small phylogenetic trees,
particularly, quartet trees. {\sc Quartet Compatibility} is the problem of
deciding whether there is a phylogenetic tree inducing a given collection of
quartet trees, and to construct such a phylogenetic tree if it exists. It is
known that {\sc Quartet Compatibility} is NP-hard and that there are only a few
results known for polynomial-time solvable subclasses. In this paper, we
introduce two novel classes of quartet systems, called complete multipartite
quartet system and full multipartite quartet system, and present
polynomial-time algorithms for {\sc Quartet Compatibility} for these systems.
