On Injective Embeddings of Tree Patterns
  We study three different kinds of embeddings of tree patterns: weakly-injective, ancestor-preserving, and lca-preserving. While each of them is often referred to as injective embedding, they form a proper hierarchy and their computational properties vary (from P to NP-complete). We present a thorough study of the complexity of the model checking problem i.e., is there an embedding of a given tree pattern in a given tree, and we investigate the impact of various restrictions imposed on the tree pattern: bound on the degree of a node, bound on the height, and type of allowed labels and edges. 
Jakub Michaliszyn, Anca Muscholl, S{\l}awek Staworko, Piotr Wieczorek   and Zhilin Wu
cs.DB