Cover-incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of the incomparability graph and the cover graph of some poset. C-I graphs captured attention as an interesting class of graphs from posets. It is known that the recognition of C-I graphs is NP-complete (Maxová et al., Order 26(3), 229–236, 2009). Hence, the problem of finding a particular graph family of C-I graphs whose recognition complexity is polynomial is interesting. We present a new forbidden subgraph characterization of Ptolemaic C-I graphs and a linear time algorithm for its recognition. The characterization of chordal C-I graph is an unsolved problem in this area for quite some time. In this paper, we characterize the family of chordal C-I graphs.

覆盖比较图（CI图）是其边集是不可比较图的边集与某些样张的覆盖图的并集的图。CI图吸引了人们的注意，因为它们是摆姿中有趣的一类图。众所周知，CI图的识别是NP完全的（Maxová等人，Order 26（3），229–236，2009）。因此，寻找识别复杂度为多项式的CI图的特定图族的问题很有趣。我们提出了一个新的禁止子图表征托勒密斯CI图和一个线性时间算法对其进行识别。弦CI图的表征在相当长的时间内是该领域尚未解决的问题。在本文中，我们描述了弦CI图的族。