On the fixed edge of planar graphs with minimum degree five (Q1917509)

An edge \(e\) of a finite simple graph \(G\) is called a fixed edge of \(G\) if \(G-e+e'\approx G\) implies \(e=e'\). The topic of fixed edges is interesting in its own right, and of particular interest to those working in graph reconstruction since certain classes of graphs can be shown to be edge reconstructible using fixed edge arguments. The main result of this paper is that a planar graph with minimum degree 5 must contain a fixed edge. The proof is quite short, but uses a result of Lauri (\textit{J. Lauri} [Edge-reconstruction of planar graphs with minimum degree 5, J. Graph Theory 3, 269-286 (1979; Zbl 0438.05048)]) about 2-connected planar graphs of minimum degree 5. The authors use their fixed edge theorem to show that planar graphs satisfying certain degree requirements are edge reconstructible. If \(G\) is a connected graph with minimum degree 1, then the trunk of \(G\), denoted \(G^\#\), is defined to be the maximal subgraph of \(G\) with minimum degree \(\geq2\). The reconstruction result in this paper states that if \(G\) is a planar graph of minimum degree 1, and if \(G^\#\) has minimum degree 5, then \(G\) is edge reconstructible. This result is a direct corollary of the main theorem on fixed edges.