On graphs determined by their \(k\)-subgraphs (Q2713604)

The following problem is solved for two graph parameters, namely the domination number and the number of edges modulo \(m\) for any \(m\geq 2\): Let \(P(G)\) be a graph parameter and let \(n,k\) and \(l\) be integers such that \(n>k>l\geq 0\). Characterize all graphs \(G\) of order \(n\) for which \(P(\langle A\rangle)=P(\langle B\rangle)\) if \(A\) and \(B\) are any \(k\)-subsets of \(V(G)\) such that \(|A\cap B|=l\). Here \(\langle D\rangle \) denotes the graph induced by the set \(D\) of vertices.