zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

Reviewer: M.M.Sysło
MSC:
05C75Structural characterization of families of graphs
06A06Partial order
68R10Graph theory in connection with computer science (including graph drawing)
68Q25Analysis of algorithms and problem complexity