Construction of all nonisomorphic minimal vertex extensions of the graph by the method of canonical representatives

In 1976 John P. Hayes proposed a graph model for investigating the fault tolerance of discrete systems. The technical system is mapped to a graph. The elements of the system correspond to the vertices of the graph, and links between the elements correspond to edges or arcs of the graph. Failure of a system element refers to the removal of the corresponding vertex from the system graph along with all its edges. Later together with Frank Harary the model was extended to links failures. The formalization of a fault-tolerant system implementation is the extension of the graph. The graph $G^*$ is called the vertex $k$-extension of the graph $G$ if after removing any $k$ vertices from the graph $G^*$ result graph contains the graph $G$. A vertex $k$-extension of the graph $G$ is called minimal if it has the least number of vertices and edges among all vertex $k$-extensions of the graph $G$. An algorithm for constructing all nonisomorphic minimal vertex $k$-extensions of the given graph using the method of canonical representatives is proposed.