Proper and unit bitolerance orders and graphs

Abstract

We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area.

Generalizations of Semiorders: A Review Note

“…Aspects of these classes are described in Langley (1993), Bogart and Isaak (1996), and Fishburn and Trotter (1997). Figure 4 illustrates split representations with intervals displaced vertically for visualization.…”

“…The purpose of this note is to bring together nine nonequivalent generalizations of semiorders, to identify all proper inclusions between their subclasses, and to point out a few other subclasses that happen to be equivalent to one of the nine. Along with the semiorder generalizations, we consider the classes of linear orders, weak orders, and bilinear orders to provide a more comprehensive view of class inclusions within P. Fishburn (1970aFishburn ( , 1985, Golumbic, Monma and Trotter (1984), Langley (1993), Bogart and Trenk (1994), Bogart, Fishburn, Isaak, and Langley (1995), and Bogart and Isaak (1996), but some are new. Other semiorder generalizations are discussed there and in Cozzens and Roberts (1982), Roubens and Vincke (1985), Doignon, Monjardet, Roubens, and Vincke (1986), Doignon (1987), Doignon, Ducamp, and Falmagne (1987), Suppes, Krantz, Luce, and Tversky (1989), Narens (1994), Mitas (1995), Abbas, Pirlot, and Vincke (1996), and Trenk (1996), among others.…”

An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy

“…Bounded multitolerance graphs (also known as bounded bitolerance graphs [3,11,15]) coincide with trapezoid graphs [11,23], which have received considerable attention in the literature, see [11]. However, the intersection model of trapezoids between two parallel lines can not cope with general multitolerance graphs, in which the set τ v of tolerance-intervals for a vertex v can be τ v = {R}.…”

Archimedean ϕ ‐tolerance graphs

“…(d) For bounded representations, the tolerance assigned to vertex v is at most the length of the interval I v ¼ ½LðvÞ; RðvÞ assigned to v. In this case, one can think of the tolerance on the leftside as a shaded sub-interval from LðvÞ to LðvÞ þ t v indicating the area of overlap permitted there without v ''complaining'' and causing an edge, and similarly, on the rightside another shaded sub-interval from RðvÞ À t v to RðvÞ indicating the overlap permitted there without causing an edge. By allowing a separate leftside tolerance and rightside tolerance for each interval, various bitolerance graph models can be obtained [3,4]. (e) Directed graph analogues to several of these models have been defined and studied [5,6,18,19].…”