  The partition lattice and noncrossing partition lattice are well studied
objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$,
its bond lattice, $L_G$, is the subposet of the partition lattice formed by
restricting to the partitions whose blocks induce connected subgraphs of $G$.
In this article, we introduce a natural noncrossing analogue of the bond
lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the
noncrossing partitions of $L_G$.
  Both the noncrossing partition lattice and the bond lattice have many nice
combinatorial properties. We show that, for several families of graphs, the
noncrossing bond poset also exhibits these properties. We present simple
necessary and sufficient conditions on the graph to ensure the noncrossing bond
poset is a lattice. Additionally, for several families of graphs, we give
combinatorial descriptions of the M\"obius function and characteristic
polynomial of the noncrossing bond poset. These descriptions are in terms of a
noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be
thought of as a noncrossing version of Whitney's NBC theorem for the chromatic
polynomial. We also consider the shellability and supersolvability of the
noncrossing bond poset, providing sufficient conditions for both. We end with
some open problems.
