On Seed Graphs with Two Components

A graph \(H\) is called a seed graph if there exists a graph \(G\) such that the deletion of any closed neighborhood of \(G\) always results in \(H\). In this paper, we investigate disconnected seed graphs. By degree and order considerations, we show that for certain pairs of connected graphs, \(H_1\) and \(H_2\), \(H_1 \cup H_2\) cannot be a seed graph. Furthermore, for every connected graph \(H\) such that \(K_1 \cup H\) is a seed graph, we show that \(H\) can be obtained by a certain graph product of \(K_2\) and \(H’\), where \(H’\) is itself a seed graph.