On \({\mathcal C}K_ 1\)-reconstruction of a pair of graphs (Q909679)

A \(K_ 1\)-concatenation \(G_ 1(u,v)G_ 2\) of graphs \(G_ 1\) and \(G_ 2\) is the graph obtained by identifying u and v, where \(u\in V(G_ 1)\) and \(v\in V(G_ 2)\). The collection \({\mathcal C}(G_ 1,G_ 2)\) is the set \(\{G_ 1(u,v)G_ 2:\quad u\in V(G_ 1),\quad v\in V(G_ 2)\}.\) A pair \(\{G_ 1,G_ 2\}\) of graphs is \({\mathcal C}K_ 1\)-reconstructible if the pair can be reconstructed from this collection, that is, if \({\mathcal C}(H_ 1,H_ 2)={\mathcal C}(G_ 1,G_ 2)\) implies \(\{H_ 1,H_ 2\}=\{G_ 1,G_ 2\}\). This paper proves that every pair of connected graphs is \({\mathcal C}K_ 1\)-reconstructible and gives a pair of disconnected graphs that is not.