## 3 Notation and Preliminaries  
This section introduces the notation and definitions used throughout the paper. First, we define an undirected labeled simple graph, since it is the type of graph the graph collection contains. **Definition 1** (Undirected labeled simple graph). : _An undirected labeled simple graph \(G\) is a five-tuple \(G=\{N,E,L,\mu,\phi\}\), where \(N\
eq\emptyset\) is the set of nodes, \(E\subset N\times N\) is the set of undirected edges, \(L\) is the set of labels for nodes and edges, \(\mu\) is the node labeling function \(\mu:N\to L\) that assigns to every node in \(N\) a label from \(L\), and \(\phi\) is the edge labeling function \(\phi:E\to L\) that assigns to every edge in \(E\) a label from \(L\)._  
From now on, whenever the term graph is used, it will refer to an undirected labeled simple graph. Next, we define graph collection, i.e., the dataset on which the proposed algorithm performs the mining process. **Definition 2** (Graph collection). : _A graph collection \(D\) is a set of \(n\) graphs denoted by \(D=\{G_{1},G_{2},\ldots,G_{n}\}\)._  
We also define a subgraph since the proposed algorithm mines frequent approximate subgraphs in a graph collection. **Definition 3** (Subgraph). : _Let \(G=\{N,E,L,\mu,\phi\}\) and \(G^{\prime}=\{N^{\prime},E^{\prime},L^{\prime},h^{\prime},\phi^{\prime}\}\) be two undirected labeled simple graphs. \(G^{\prime}\) is a subgraph of \(G\) denoted by \(G^{\prime}\subseteq G\) if the following conditions are accomplished: \(N^{\prime}\subseteq N\), i.e., the nodes of \(G^{\prime}\) are a subset of the nodes of \(G\), and \(E^{\prime}\subseteq E\), i.e., the edges of \(G^{\prime}\) are a subset of the edges of \(G\), \(\forall\ u\in V^{\prime}:\mu^{\prime}(u)=\mu(f(u))\) such that \(\mu^{\prime}(u)\in L^{\prime}\) and \(\mu(f(u))\in L\), \(\forall\ (u,v)\in E^{\prime}:(f(u),f(v))\in E\), and \(\phi^{\prime}(u,v)=\phi(f(u),f(v))\) such that \(\phi^{\prime}(u,v)\in L^{\prime}\) and \(\phi(f(u),f(v))\in L\)._  
An important part of frequent approximate subgraph mining is determining the similarity between graphs. Therefore, a graph similarity measure is required to determine how similar the subgraphs can be. The dissimilarity function used is based on the graph edit distance [63], which consists of counting the operations required to transform one graph into another. **Definition 4** (Dissimilarity function). : _Let \(G_{1}=(N_{1},E_{1},L_{1},\mu_{1},\phi_{1})\) and \(G_{2}=(N_{2},E_{2},L_{2},\mu_{2},\phi_{2})\) be two graphs, \(\gamma\) be a node label similarity, and \(\delta\) be an edge label similarity. The dissimilarity function to be used in the proposed algorithm is the sum of all the costs required to transform a graph \(G_{1}\) into a graph \(G_{2}\). Therefore, the dissimilarity of transforming \(G_{1}\) to \(G_{2}\) is given by:_  
\[\text{dissimilarity}(G_{1},G_{2})=\sum_{i=1}^{k}CN_{i}+\sum_{j=1}^{m}CE_{j} \tag{1}\]  
_where \(CN_{i}\) represents the sum of the \(k\) modification costs for the nodes, and \(CE_{j}\) represents the sum of the \(m\) modification costs for the edges. (Note: Depending on the specific structure of the graphs, the node modification cost \(k\) or the edge modification cost \(m\) can be zero. )_  
The possible modification costs are:  
Substitution of labels on a node or an edge. * The node label substitution cost is \(1-\gamma(v_{1},v_{2})\), where \(\gamma(v_{1},v_{2})\in[0,1]\) represents the similarity between the label of the node \(v_{1}\) and the label of the node \(v_{2}\). * The edge label substitution cost is \(1-\delta(\epsilon_{1},\epsilon_{2})\), where \(\delta(\epsilon_{1},\epsilon_{2})\in[0,1]\) represents the similarity between the label of the edge \(\epsilon_{1}\) and the label of the edge \(\epsilon_{2}\). Insertion of a node or an edge. * The node insertion cost of inserting a \(v_{1}\)-labeled node is \(1\). * The edge insertion cost of inserting an \(\epsilon_{1}\)-labeled edge is \(1\). Deletion of a node or an edge. * The node deletion cost of deleting a \(v_{2}\)-labeled node is \(1\).