On k-tree Containment Graphs of Paths in a Tree

A k -tree is either a complete graph on k vertices or a graph that contains a vertex whose neighborhood induces a complete graph on k vertices and whose removal results in a k -tree. If the comparability graph of a poset P is a k -tree, we say that P is a k -tree poset. In the present work, we study and characterize by forbidden subposets the k -tree posets that admit a containment model mapping vertices into paths of a tree ( CPT k -tree posets). Furthermore, we characterize the dually- CPT and strong- CPT k -tree posets and their comparability graphs. The characterizations lead to efficient recognition algorithms for the respective classes.