  A finite simple graph $\Gamma$ determines a quotient $P_\Gamma$ of the pure
braid group, called a graphic arrangement group. We analyze homomorphisms of
these groups defined by deletion of sets of vertices, using methods developed
in prior joint work with R. Randell. We show that, for a $K_4$-free graph
$\Gamma$, a product of deletion maps is injective, embedding $P_\Gamma$ in a
product of free groups. Then $P_\Gamma$ is residually free, torsion-free,
residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex.
We also show $P_\Gamma$ is of homological finiteness type $F_{m-1}$, but not
$F_m$, where $m$ is the number of copies of $K_3$ in $\Gamma$, except in
trivial cases. The embedding result is extended to graphs whose 4-cliques share
at most one edge, giving an injection of $P_\Gamma$ into the product of pure
braid groups corresponding to maximal cliques of $\Gamma$. We give examples
showing that this map may inject in more general circumstances. We define the
graphic braid group $B_\Gamma$ as a natural extension of $P_\Gamma$ by the
automorphism group of $\Gamma$, and extend our homological finiteness result to
these groups.
