  We introduce a variation of interval graphs, called veto interval (VI)
graphs. A VI graph is represented by a set of closed intervals, each containing
a point called a veto mark. The edge $ab$ is in the graph if the intervals
corresponding to the vertices $a$ and $b$ intersect, and neither contains the
veto mark of the other. We find families of graphs which are VI graphs, and
prove results towards characterizing the maximum chromatic number of a VI
graph. We define and prove similar results about several related graph
families, including unit VI graphs, midpoint unit VI (MUVI) graphs, and single
and double approval graphs. We also highlight a relationship between approval
graphs and a family of tolerance graphs.
