Metric for comparing graphs with ordered vertices based on the maximum common subgraph

The paper is devoted to the methods of comparison and classification of graphs. This direction is known as graph matching. An overview of metrics for comparing graphs based on a maximum common subgraph is given. A graph $\text{mcs}(G, F)$ is a maximal common subgraph of graphs $G$ and $F$ if it is isomorphic to $G' \subseteq G$ and $F' \subseteq F$ and contains the maximum number of vertices. In some tasks (for example, comparing texts), it is important to take into account one more factor: vertex numbering. A modification of the distance based on the maximum common subgraph is proposed, taking into account this factor (each vertex has its own unique number). We determine a function of graphs $G$ and $F$ as follows: $d(G, F) = 1 - \min_{i=1,\dots,k}({|\text{mcs}(g_{\min(i,m)}, f_i)|}/{i})$. Here $|G|$ denotes the number of vertices in $G$, $|G|=m$, $|F|=k$, $m\leq k$; and $g_i$ is the subgraph of $G$ containing vertices with numbers from $1$ to $i$ and all edges of $G$ incident to these vertices (the graphs $f_i$ are defined similarly). It is shown that this function satisfies all the properties of the metric (nonnegativity, identity, symmetry, triangle inequality). This metric can be used to solve various problems of image recognition (for example, to establish the authorship of texts).