3
$\begingroup$
  1. Let $L(G)$ be the line graph of a graph $G$. Is $G$ the line graph of $L(G)$?
  2. 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.

  3. Let $D(H)$ be the dual graph of a planar graph $H$. Is $H$ the dual graph of $D(H)$?
  4. Are there results for dual graphs similar to part 2 for line graphs?

Thanks!

  • 0
    "Translated" is not a mathematical term. One underlying issue is that any two true logical statements are equivalent, and so one can be "translated" into the other. But the translation is probably not going to be very useful.2012-10-30

0 Answers 0