The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations.We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. This includes the case of partial representation extension.The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs [Balko et al. ISAAC'13]. So unless P = NP, proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.

Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

Abstract. Interval graphs are intersection graphs of closed intervals of the real-line. The wellknown computational problem, called recognition, asks whether an input graph G can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker [J. Comput. System Sci., 13 (1976)] based on PQ-trees.In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph G with a partial representation R ′ fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation R of the entire graph G extending R ′ . We generalize the characterization of interval graphs by Fulkerson and Gross [Pac. J. Math., 15 (1965)] to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.1. Introduction. One of the fundamental themes of mathematics is studying relations between mathematical objects and their representations. For graph theory, the study of graph representations and graph drawing is as old as the study of graphs themselves. A widely studied type of graph representations are intersection representation which encode edges by intersections of sets. An intersection representation R of a graph G assigns a collection of sets R v | v ∈ V (G) such that uv ∈ E(G) if and only if R u ∩ R v = ∅. Since every graph has an intersection representation [23], interesting graph classes are obtained by restricting the representing sets to some nice class of, say, geometrical objects, e.g., continuous curves in plane, chords of a circle, convex sets, etc. For overview of these classes, see books [10,25,31].The most famous are interval graphs (INT) which are intersection graphs of closed intervals of the real line. It is one of the oldest classes of graphs, introduced by Hajós [11] already in 1957. Interval graphs have many useful theoretical properties, for example they are perfect and related to path decompositions. In many cases, very hard combinatorial problems are polynomially solvable for interval graphs [30]; e.g., maximum clique, k-coloring, maximum independent set, etc. Also, interval graphs naturally appear in many applications concerning biology, psychology, time scheduling, and archaeology; see for example [29,32,3].

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