Graphs with cyclomatic number two having panconnected square

The square   of a graph GG is the graph obtained from GG by adding edges joining those pairs of vertices whose distance from each other in GG is two. If GG is connected, then the cyclomatic number   of GG is defined as |E(G)|−|V(G)|+1|E(G)|−|V(G)|+1. Graphs with cyclomatic number no more than one whose square is Hamilton-connected have been characterized, among other things, in Chia et al. (2009) [3]. Here, we characterize all graphs GG with cyclomatic number two whose square is Hamilton-connected.