  Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs
from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set
$V$, and the edge-set is the union of edge sets of the cover graph and the
incomparability graph of the poset. The recognition of the C-I graphs is known
to be NP-complete (Maxov\'{a} et al., Order 26(3), 229--236(2009)). In this
paper, we prove that chordal graphs having at most two independent simplicial
vertices are exactly the chordal graphs which are also C-I graphs. A similar
result is obtained for cographs as well. Using the structural results of these
graphs, we derive linear time recognition algorithms for chordal graphs and
cographs which are C-I graphs.
