Algorithmic aspects of comparability graphs and interval graphs.

*(English)* Zbl 0569.05046
Graphs and order. The role of graphs in the theory of ordered sets and its applications, Proc. NATO Adv. Study Inst., Banff/Can. 1984, NATO ASI Ser., Ser. C 147, 41-101 (1985).

[For the entire collection see

Zbl 0549.00002.]
Comparability graphs are undirected graphs that represent the comparability relation of partial orders. They constitute an important interface between graphs and partial orders both for theoretical investigations and the development of coefficient algorithms for otherwise NP-hard problems on partial orders and their comparability graphs.
The first part of this paper is a survey of this second aspect of comparability graphs. Presented are algorithmic methods and the necessary theoretical background for comparability graph recognition, for constructing all partial orders with the same comparability graph, for decomposing comparability graphs and partial orders, for determining comparability invariants such as order dimension or jump number by decomposition, and for solving combinatorial optimization problems on comparability graphs.
The second part deals with the class of interval graphs which are the incomparability graphs of interval orders. Again, algorithmic methods for interval graph recognition and for solving combinatorial optimization problems on these graphs are discussed.
Finally, it is demonstrated how comparability graphs and interval graphs can be used for solving the seriation problem in archeology and some scheduling problems on partial orders.

##### MSC:

05C75 | Structural characterization of families of graphs |

06A06 | Partial order |

68R10 | Graph theory in connection with computer science (including graph drawing) |

68Q25 | Analysis of algorithms and problem complexity |