I recently tried to prove the following characterization of [chordal graphs](http://en.wikipedia.org/wiki/Chordal_graph), attributed to Fulkerson & Gross:


"A graph $G$ is chordal if and only if it has an ordering such that for all $v \in G$, all the neighbors of $v$ that precede it in the ordering form a clique."


I believe the wikipedia page calls this (or its reverse) a "perfect elimination ordering." In any case, my proof was harder than I expected and I ended up with the following strengthening: 


"$G$ is chordal if and only if for any $v \in G$, we can find such an ordering of $G$ that starts with $v$."


I would like to know if this strengthening is:


1) known / obvious from the weaker theorem / obvious from folklore, or
2) wrong. =)


I can of course give a proof to anyone who is interested - it is just too long to fit here.


Best,
-Yan



characterization of chordal graphs
It's known/obvious from the next few lines of the Wikipedia article stating that a LexBFS ordering (or its reverse, depending on your conventions) gives a perfect elimination ordering, since LexBFS can be made to start at any vertex.


