- Let $L(G)$ be the line graph of a graph $G$. Is $G$ the line graph of $L(G)$?
From the same article:
Properties of a graph $G$ that depend only on adjacency between edges may be translated into equivalent properties in $L(G)$ that depend on adjacency between vertices.
Are the followings also true?
Properties of $L(G)$ that depend only on adjacency between vertices may be translated into equivalent properties in $G$ that depend on adjacency between edges*.
properties of a graph $G$ that depend only on adjacency between vertices may be translated into equivalent properties in $L(G)$ that depend on adjacency between edges.
properties of a graph $L(G)$ that depend only on adjacency between edges may be translated into equivalent properties in $G$ that depend on adjacency between vertices.
- Let $D(H)$ be the dual graph of a planar graph $H$. Is $H$ the dual graph of $D(H)$?
- Are there results for dual graphs similar to part 2 for line graphs?
Thanks!