Abstract
AbstractTheinterval count problemdetermines the smallest number of interval lengths needed in order to represent an interval model of a given interval graph or interval order. Despite the large number of studies about interval graphs and interval orders, surprisingly only a few results on the interval count problem are known. In this work, we provide a short survey about the interval count and related problems. a graph and the number of its maximal cliques.
Highlights
The interest in interval graphs and orders comes from both their central role in many applications and purely theoretical questions [15, 17, 29]
The outputs of interest among the applications that require the construction of interval models can be constrained by a desirable number of distinct interval lengths, generally as a matter of convenience or fairness
The existence of a set S of simplicial vertices such that G[V (G) \ S] is a unit interval graph does not suffice to state that there is a model of G having only zero or one interval length, as exemplified in [30]
Summary
The interest in interval graphs and orders comes from both their central role in many applications and purely theoretical questions [15, 17, 29]. The minimum number of lengths required for any interval model of this graph is two, which is, the interval count of this graph. The outputs of interest among the applications that require the construction of interval models can be constrained by a desirable number of distinct interval lengths, generally as a matter of convenience or fairness. The interval graph to be generated as a result of the scheduling process is known to have a certain interval count by construction (which is, the total number of distinct required class lengths). Since the interval count problem is defined only for interval graphs Denoting by IC(R) the number of distinct lengths of a given interval model R, for a given order P , we can write. Note that when a result is presented only for the interval count of graphs For the omitted notation in this paper, refer to [5] for general graph theory, [32] for general order theory, and [15, 17] for a specialized discussion about interval graphs and interval orders
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