Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter II: algorithms

Abstract

This is the second and last chapter of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version (Rep), a proper circular-arc (PCA) model M is given and the goal is to obtain an equivalent UCA model U. In the bounded version (BoundRep), M is given together with some lower and upper bounds that the beginning points of U must satisfy. In the minimal version (MinUCA), the circumference of the circle and the length of the arcs in U must be simultaneously as small as possible, while the separation of the extremes is greater than a given threshold. In this chapter we take advantage of the theoretical framework developed in Chapter I to design efficient algorithms for these problems. We show a linear-time algorithm with negative certification for Rep, that can also be implemented to run in logspace. We develop algorithms for different versions of BoundRep that run in linear space and quadratic time. Regarding MinUCA, we first show that the previous linear-time algorithm for MinUIG (i.e., Min-UCA on UIG models) fails to provide a minimal model for some input graphs. We fix this algorithm but, unfortunately, it runs in linear space and quadratic time. Then, we apply the algorithms for MinUIG and MinUCA (Chapter I) to find the minimum powers of paths and cycles that contain given UIG and UCA models, respectively.

Extending partial representations of circle graphs

“…In the context of intersection-defined classes, this problem was first considered in [29] for interval graphs. Currently, the best known results are linear-time algorithms for interval graphs [5,28] and proper interval graphs [26], a quadratic-time algorithm for unit interval graphs [26,37,38], and polynomial-time algorithms for permutation and function graphs [25], proper circular-arc graphs [3], and trapezoid graphs [32]. For chordal graphs (as subtree-in-a-tree graphs) several versions of the problems were considered [27] and all of them are NP-complete, and similarly for different contact representations of planar graphs [9].…”

“…This problem is clearly a generalization of partial representation extension, since one can convert an instance of the partial representation extension problem into an instance of the bounded representation problem by replacing every occurrence of a vertex by a small circular arc (such that no two circular arcs so introduced intersect) and prescribing a circular arc that covers the whole circle for every endpoint that does not appear in the instance. It is known to be polynomially solvable for interval and proper interval representations of interval graphs , and surprisingly it is ‐complete for unit interval representations . The complexity for other classes is not known.…”

On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

“…Minimal obstructions making partial representations non-extendible are described in [23]. A linear-time algorithm for proper interval graphs [17] and a quadratic-time algorithm for unit interval graphs [31] are known.…”

Minimal Obstructions for Partial Representations of Interval Graphs

“…We note that the partial representation extension problems have been considered also for other classes of intersection graphs. algorithm for unit interval graphs are given in [19], and improved to quadratic time in [33]. The partial representation extension problems are polynomial-time solvable for k-nested interval graphs (classes generalizing proper interval graphs), but NP-hard for k-length interval graphs (classes generalizing unit interval graphs), even for k = 2 [23].…”