  We propose a novel way of generalizing the class of interval graphs, via a
graph width parameter called the simultaneous interval number. This parameter
is related to the simultaneous representation problem for interval graphs and
defined as the smallest number $d$ of labels such that the graph admits a
$d$-simultaneous interval representation, that is, an assignment of intervals
and label sets to the vertices such that two vertices are adjacent if and only
if the corresponding intervals, as well as their label sets, intersect. We show
that this parameter is $\mathsf{NP}$-hard to compute and give several bounds
for the parameter, showing in particular that it is sandwiched between
pathwidth and linear mim-width. For classes of graphs with bounded parameter
values, assuming that the graph is equipped with a simultaneous interval
representation with a constant number of labels, we give $\mathsf{FPT}$
algorithms for the clique, independent set, and dominating set problems, and
hardness results for the independent dominating set and coloring problems. The
$\mathsf{FPT}$ results for independent set and dominating set are for the
simultaneous interval number plus solution size. In contrast, both problems are
known to be $\mathsf{W}[1]$-hard for linear mim-width plus solution size.
