There are two relevant questions:
(1) We know an edge set $C$ is a rigid cycle in $\mathcal{G}_2(n)$ if and only if $|E(C)|=2|V(C)|-2$ and $|F|\leq 2|V(F)|-3$ for every proper subset $F$ of $E(C)$. Thus, I want to know: Is a rigid cycle a chordal graph? (It can be found in Wiki that chordal graphs are also called as "rigid circuit graphs". But i can't find the connection between it and rigid cycle. Are they the same thing?)
(2) A graph $G=(V,E)$ is defined as pseudocycle iff $|E|=2|V|-2$ and $F\leq 2|V(F)|-2$, $\forall\ \emptyset\subset F\subset E$. Thus, I also would like to know: Is a pseudocycle a chordal graph?
Thanks very much!
Construct a graph $G$ by starting with a $C_4$ (cycle of length 4) on vertices $a,b,c,d$, then adding a fifth vertex $e$ adjactent to all other four vertices.
Notice that $G$ isn't chordal because of the $C_4$.
Now $|E(G)| = 8 = 2|V(G)| - 2$.
Also, take any $F \subset E(G)$. If $|V(F)| = 5$, then clearly the inequality is satisfied.
Otherwise say $|V(F)| = 4$. Then either $V(F) = \{a,b,c,d\}$ and $|F| \leq 4 \leq 2|V(F)| - 3$, or $V(F)$ is $e$ and 3 other vertices and $|E(F)| \leq 5 \leq 2|V(F)| - 3$. You can verify the case with $|V(F)| = 3$.