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**Extra resources for Categorical foundations: special topics in order, topology, algebra, and Sheaf theory**

**Example text**

The following are equivalent statements about a graph G. (i) G is a unit tolerance graph. (ii) G is a 50% tolerance graph. (iii) G has a bounded tolerance representation with constant cores. The proof of (i) ⇐⇒ (ii) appears in Langley (1993) and Bogart, Fishburn, Isaak, and Langley (1995). 26. 26. As we discuss other types of tolerance graphs, we will revisit the question of whether the unit and proper classes are equal or not. We conclude this section with the result due to Bogart, Jacobson, Langley, and McMorris (2001) showing that interval graphs are not just tolerance graphs, but unit tolerance graphs.

6. 16. A concatenation of three permutation diagrams, its intersection graph G and a transitive orientation F of the complement G. 9. The orientation F is obviously transitive, so G is a comparability graph. (iii) =⇒ (iv): Let G be the comparability graph of an order P = (X, ≺), and let L = {L 1 , L 2 , . . , L k+1 } be a realizer of P. We may assume, without loss of generality, that X = {1, 2, . . , n}. We will build a concatenation of permutation diagrams whose intersection graph will be G.

An interval containment graph is one that can be represented by a set of real intervals {Ii | i ∈ V (G)} so that i j ∈ E(G) precisely when one of Ii ,I j contains the other. Such a representation is called an interval containment representation. 1 gives such a representation for the complete bipartite graph K 3,3 . The equivalence of (ii) and (iii) in the following theorem first appeared in Dushnik and Miller (1941). 6. The following are equivalent statements about a graph G. (i) (ii) (iii) (vi) G G G G has a tolerance representation with ti = |Ii | for all i ∈ V (G).