## 1 Introduction  
Graphs have been used to model many complex data objects and their relationships in our real world, such as bioinformatics, chemistry, social networks, software, the World Wide Web, and so on [1; 2; 3; 4; 5; 6]. With the increasing of graph data, graphs have been of extensive concern and deeply studied. The management and mining of graph data can effectively solve the analysis and processing problems of topological data [7; 8; 9]. One of the most important tasks in graph management is how to effectively answer the various queries in graph databases. For example, medical staff needs to query whether a given compound contains a particular substructure, and scientists want to query the number of specific substructures such as triangle subgraphs in the graph database.  
In this paper, a novel type of interesting subgraph query is proposed. Suppose you are searching in the human protein network using query position \(q\) (a specific gene at a specific position), and you want to find a suitable subgraph containing the query position in the network for visualization purpose. The subgraph should not be too small to avoid users not being able to distinguish the differences between the neighborhood of the query position and that of other vertices, and the subgraph should not be too large to avoid users getting confused with the returned redundant information. The subgraph should be special and representative. For this demand, this paper presents the Minimal Unique Induced Subgraph (MUIS) query.  
MUIS query refers to finding out a unique induced subgraph that contains a minimum number of vertices and the given query position. MUIS query provides a new graph data access and management method that has many potential applications, such as subgraph retrieval, graph visualization, representative subgraph discovery and vertex property exploration.  
Answering MUIS query efficiently is far from trivial. There are many academic problems in the research, such as what are the properties of MUIS, how to find MUIS candidate subgraphs, how to prune useless candidate subgraphs as early as possible, how to ensure the minimum number of vertices of MUIS and how to check the uniqueness of candidate graphs.  
We summarize the major contributions by the following four ingredients:  
* To our best knowledge, we are the first to propose MUIS query, which is a novel type of interesting and useful subgraph query. MUIS query enriches and develops graph data query and management methods;
* For the novel type of subgraph query, the formal definition is given and the properties are discussed in this paper;
* The EQA (Efficient Query Answering) algorithm is proposed to solve the MUIS query problem under the filtering-validation framework. In the EQA algorithm, BFS (Breadth First Search)-based candidate set generation strategy, matched vertices-based pruning strategy and query position-based subgraph isomorphism are proposed to improve the effectiveness and efficiency of MUIS query;
* Through comprehensive experiments on real datasets and synthetic datasets, EQA is demonstrated to outperform the state-of-the-art model to answer MUIS query. Influencing factors of the process speed are also verified by the experiments.  
The rest of the paper is organized as follows. We define MUIS and discuss the properties in Section 2, and review the related work briefly in Section 3. Then, we present our model in Section 4. In Section 5, we report the experimental results, and we conclude the paper in Section 6.