Treelike Comparability Graphs

A comparability graph is a simple graph which admits a transitive orientation on its edges. Each one of such orientations defines a poset on the vertex set, and also it is said that this graph is the comparability graph of this poset. A treelike poset is a poset whose covering graph is a tree. ability graphs of arborescence posets are known as trivially perfect graphs. These have been characterized by Wolk and Golumbic. s article we study treelike comparability graphs, that is, comparability graphs of treelike posets. We prove necessary and sufficient conditions that a prime comparability graph must verify for being a treelike comparability graph. Based on the modular decomposition we give a characterization of treelike graphs.