are *parallel* if $G-F$ has at most one connected component $H$ where $E(H) \cap F' \neq \emptyset$.

A *chord* is an edge between two nonadjacent vertices of a cycle. A graph $H$ is *chordal* if and only if every cycle of length four or greater in $H$ has a chord. A chordal graph $H$ is a *triangulation* of graph $G$ if and only if $V(G) = V(H)$ and $E(G) \subseteq E(H)$. The edges in $E(H) \setminus E(G)$ are called *fill-in* edges of $G$. A *clique tree* of a chordal graph $H$ is a pair $(T, B)$ where (i) $T$ is a tree, (ii) $B$ is a bijective function from vertices of $T$ to maximal cliques of $H$, and (iii) $(T, B)$ satisfies the *coherence* property; i.e., for every vertex $v \in H$, the set of all vertices $x$ of $T$ where $v \in B(x)$ induces a subtree in $T$.

Let $G$ be a graph and let ${\ensuremath{\mathcal{F}}}$ be a collection of subsets of $V(G)$. We represent by $G_{{\ensuremath{\mathcal{F}}}}$ the graph derived from $G$ by making the set of vertices of $X$ a clique in $G$ for every $X \in {\ensuremath{\mathcal{F}}}$.

\[thm:parallel\_minimal\_ct\] Let ${\ensuremath{\mathcal{F}}}$ be a maximal set of pairwise parallel 