Powers of distance-hereditary graphs

AbstractPowers of distance-hereditary graphs need not be distance-hereditary, but they come close: the house, the domino and cycles of length larger than four are forbidden induced subgraphs, whereas the 5-vertex fan may occur. Moreover, all even powers are chordal, and an odd power is chordal if and only if the given distance-hereditary graph has no induced subgraph of a certain type (viz. a 4-cycle with pendant paths of suitable length). In addition, one can exhibit further small graphs that are forbidden (as induced subgraphs) in either all or just some powers of distance-hereditary graphs. Other invariants of these powers can be expressed in terms of distances (yielding a 4-point condition) or intersection patterns (viz. Helly type conditions) of disks