On seed graphs with two components (Q2713614)

A graph \(H\) is called a seed graph, if there exists a graph \(G\) such that for each vertex \(v\) of \(G\) the subgraph of \(G\) induced by the set of all vertices having distance at least 2 from \(v\) is isomorphic to \(H\). The paper studies seed graphs with two connected components. Some examples of pairs of connected graphs \(H_1\), \(H_2\) are shown such that both \(H_1\) and \(H_2\) are seed graphs, while their disjoint union \(H_1\cup H_2\) is not. Some assertions are proved giving conditions for two connected graphs in order that their disjoint union might be a seed graph.