Abstract
The comparability graph G of an ordered set P = <X;≤ > is the graph with vertex set X and edges xy whenever x < y or x > y in P. We discuss some of the important results about comparability graphs. We ignore algorithmic aspects of comparability graphs because they are the subject of Möhring [1985].
The characterization problem for comparability graphs was considered first. The same characterization is given in Ghouila-Houri [1962] and in Gilmore and Hoffman [1964]. The next major result was T. Gallai’s [1967] characterization of finite comparability graphs in terms of the minimum list of graphs that are excluded as induced subgraphs. This paper contains a penetrating analysis of finite comparability graphs. In particular, Gallai [1967] obtained all orderings of a finite comparability graph in such a wav that they are easily counted. L.N. Shevrin and N.D. Filippov [1970] later did the same thing for infinite comparability graphs. N.D. Filippov [1968a, 1968b] continued this investigation of comparability graphs. We extend Gallai’s ideas to the infinite case.
B. Dushnik and E.W. Miller [1941] defined the dimension of an ordered set to be the minimum cardinality of a realizer. The dimension is a comparability invariant, that is, it has the same value on all ordered sets with the same comparability graph. For the finite case, this was proved independently in Gysin [1976] and Trotter, Moore and Sumner [1976J. The general case was proved by J.C. Arditti and H.A. Jung [1980]. M. Habib [1984] determined many other invariants of finite comparability graphs. We extend Gallai’s ideas to the infinite case.
B. Dushnik and E.W. Miller [1941] defined the dimension of an ordered set to be the minimum cardinality of a realizer. The dimension is a comparability invariant, that is, it has the same value on all ordered sets with the same comparability graph. For the finite case, this was proved independently in Gysin [1976] and Trotter, Moore and Sumner [1976]. The general case was proved by J.C. Arditti and H.A. Jung [1980]. M. Habib [1984] determined many other invariants of finite comparability graphs.
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Kelly, D. (1985). Comparability Graphs. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_1
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