I am trying to prove a result about rigid graphs, which I believe holds for chordal graphs and also non-chordal but $M$-connected graphs (note: when I say $M$-connected I am referring to the rigidity matroid). Clearly, by rigidity, all graphs I mention will be at least 2-connected.
So...I need to pinpoint exactly what special properties an $M$-connected but NOT chordal graph has, which a chordal graph does not. Is there some kind of characterisation of these?
In particular, I believe I'm right in saying that there exist $M$-connected graphs having 2-separations (by which I mean pairs of vertices whose deletion disconnects the graph) ${u,v}\subseteq V(G)$ which are such that $uv\notin E(G)$. (As opposed to chordal graphs, in which every 2-separating pair is adjacent).
Unfortunately I don't seem to be able to come up with any examples! I can't seem to find an $M$-connected graph having a 4-cycle (please excuse my ignorance if this is obvious---I am very new to the concept of $M$-connectedness, and don't find it intuitive).
I would be interested to hear of any other difference between graphs which are chordal, and those which are $M$-connected and not chordal. With regards to rigidity, both types have the property that any equivalent realisation can be obtained by iteratively reflecting components around 2-separations. Chordal graphs consist of $K_2$-sums of 3-connected graphs. Is there an analagous characterisation of $M$-connected graphs?
Finally, for generic realisations $(G,p)$ of graphs $G$, if there is anyone who is familiar with the concept of the transcendental field extension $\mathbb Q(d_G(p))$ of the rationals formed by adding all edge-distance equations defining $G$ to $\mathbb Q$...is it true that adding an edge between a non-adjacent 2-separating pair ${u,v}$ does not change $\mathbb Q(d_G(p))$? ie. is it true that $d_(u,v)\in \mathbb Q(d_G(p))$ for every 2-separation ${u,v}$??
Thanks! (and sorry for the rambling question)
