For a graph G of order n and an integer k with \(1\le k\le n\) , let \(\pi (G,k)\) denote the minimum number of edges spanned by k vertices and call \((\pi (G,1),\pi (G,2),\ldots ,\pi (G,n))\) the sparse sequence of G. In 2022, Katona proposed a problem: Can we find necessary and sufficient conditions for a function f(k) under which a graph G exists such that \(\pi (G,k)=f(k)\) ? In this paper, we solve this problem partly and give a sufficient condition for a sequence to be the sparse sequences of a general graph and a tree, respectively. For any tree T and \(1\le k\le n-2\) , we show that \(\pi (T,k+2)-\pi (T,k+1)\ge \pi (T,k+1)-\pi (T,k)-1\) . Finally, we introduce the subgraph-size polynomial of a graph and establish a recursive relation for graphs with a cut edge, based on which we give a recursive algorithm for determining the sparse sequences of a tree and obtain the subgraph-size polynomial of spider graphs.
