Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these substructures. However, most of the existing graph kernels do not have the property of scale-adaptivity, i.e., they cannot compare graphs at multiple levels of granularities. Many real-world graphs such as molecules exhibit structure at varying levels of granularities. To tackle this problem, we propose a new graph kernel called Tree++ in this paper. At the heart of Tree++ is a graph kernel called the path-pattern graph kernel. The path-pattern graph kernel first builds a truncated BFS tree rooted at each vertex and then uses paths from the root to every vertex in the truncated BFS tree as features to represent graphs. The path-pattern graph kernel can only capture graph similarity at fine granularities. In order to capture graph similarity at coarse granularities, we incorporate a new concept called super path into it. The super path contains truncated BFS trees rooted at the vertices in a path. Our evaluation on a variety of real-world graphs demonstrates that Tree++ achieves the best classification accuracy compared with previous graph kernels.

Three-way concept analysis in incomplete contexts lays the theory dealing with the data in incomplete contexts, especially three kinds of partially known formal concepts including SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept. Generally speaking, not every attribute is essential in an incomplete context since the purpose of research is different. Thus, we propose four kinds of attribute reduction of SE-ISI concept lattices based on different criteria. Then, we discuss the relationships among the four kinds of attribute reduction, including the relationships among the consistent sets and relationships among the reducts. Finally, based on discernibility matrices and discernibility functions, the approaches to obtaining these attribute reduction are presented. Keywords Incomplete context • Concept lattice • SE-ISI formal concept • Attribute reduction • Discernibility matrix 1 Introduction Formal concept analysis (FCA), an efficient tool for decision making and knowledge discovery, was proposed by Wille (1982) and Ganter and Wille (1999). Formal context, formal concept and concept lattice are three basic notions of FCA. Formal context is the data foundation of FCA. Based on a formal context, formal concept, a pair of extent and intent, Communicated by A. Di Nola.

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