For a graph G and a subset A of its vertices denote by Bdry(A) the set of vertices of the complement that are at distance 1 from A. Furthermore, denote by Ball(A) the ball around A, i.e. the union of A and Bdry(A).
This tool constructs a graph G corresponding to the given ball-sequence. The i-th element of such sequence is the minimum size of the ball around a set of size i, for i=1, 2, ... , |G|. We assume here that the graph is connected. This means that the ball-sequence is not decreasing, and the i-th entry is at least min(i+1, |G|), for i=1, 2, ... , |G|.
The ball-sequence does not specify the graph uniquely. Moreover, not any ball-sequence satisfying the above properties is realized by some graph. However, any such sequence is realized by an oriented graph. In this case the boundary is defined as the set of vertices that can be reached from A by outgoing edges. The oriented graph is obtained by connecting the vertex i with the vertices {j < i : |Ball(j)| > i-1}.
Enter the ball-sequence into the text input field (separate its entries by spaces) and click the "Check it" button. The corresponding graph will be displayed. The edges connecting vertices within a level are omitted. The orientation of edges is not shown. The adjacency list of the constructed graph will be shown in a separate tab.
Last modified: Mon, Jan 23, 2023.