## 2 Related Work
### Frequent Approximate Subgraphs in a Graph Collection  
Graph mining algorithms reported in the literature have addressed three problems in a graph collection: only structural variations, only label variations, and a combination of label and structural variations. RAM [58] and UMFGAMW [59] algorithms address structural variations in a graph collection. RAM allows for structural variations in the edges by allowing their absence. It does not allow label variations, and it does not allow structural variations in the nodes. It does not mine all the frequent subgraphs because it performs a random selection of candidates, eliminating those with similar characteristics to candidates that have already been analyzed. UMFGAMW [59] mines uncertain maximal frequent subgraphs in a graph collection. Due to the nature of uncertain graphs, it mines approximate subgraphs with a similar structure based on the probability of an edge's presence. APGM [42], and VEAM [41] have addressed label variation. APGM mines clique frequent subgraphs by representing the graph using a canonical adjacency matrix (CAM). To handle the label variations, it uses a node probability substitution matrix. VEAM mines all frequent subgraphs with label variations in nodes and edges. It uses two probability substitution matrices. Algorithms such as AMgMiner [60], MgVEAM [61], and CliqueAMgMiner [62] have been reported in the literature for frequent subgraph mining in a graph collection of multigraphs with label variations. AMgMiner preprocesses the multigraphs to convert them into canonical adjacency matrix codes. It occupies two probability substitution matrices to allow for label variations in nodes and edges. MgVEAM, unlike the previous one, introduces a method to mine multigraphs without prior preprocessing by creating a canonical adjacency matrix code that takes multigraphs into account. CliqueAMgMiner mines the clique subgraphs of a multigraph collection. Regarding label and structural variations in a graph collection, the REAFUM [48] framework allows label and structural variations in nodes and edges. It consists of three stages: selection of representative graphs, expansion of the frequent nodes in search of approximate frequent subgraphs, taking into account the variations in label and structure, and finally, a consensual refinement. To the best of our knowledge, REAFUM is the only algorithm reported in the literature for mining frequent subgraphs in a graph collection. It allows label and structural variations in nodes and edges. However, it does not mine all the frequent subgraphs in a collection; instead, it mines them just from a subset of representative subgraphs, nor does it allow partial similarity in labels. Table 1 shows that allowing structural variations for mining frequent approximate subgraphs in graph collections has been scarcely studied. This problem has been explored a bit more for mining frequent approximate subgraphs in a single graph, with a few options reported in the literature. Even though there is an algorithm called REAFUM reported in the literature for mining frequent subgraphs in graph collections considering label and structural variations, it does not mine all frequent approximate subgraphs; it only mines some representative frequent subgraphs. To the best of our knowledge, no algorithm can mine all frequent approximate subgraphs in a graph collection considering both structural and label variations. Therefore, as shown in Table 1, we propose AGCM-SLV.