Abstract. Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to swap the tokens on adjacent vertices. We prove that every such puzzle is solvable in O(n 2 ) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement. We give a polynomial-time 2-approximation algorithm for trees, and using this, obtain a polynomial-time 2α-approximation algorithm for graphs whose tree α-spanners can be computed in polynomial time. Finally, we show that the problem can be solved exactly in polynomial time on complete bipartite graphs.

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.

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].

Chordal graphs are intersection graphs of subtrees of a tree T . We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T ′ and some pre-drawn subtrees of T ′ . It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e, keeps the pre-drawn subtrees unchanged).We consider four modifications of T ′ and get vastly different problems. In some cases, it is interesting to consider the complexity even if just T ′ is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization.We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components and the size of the tree T ′ . We describe an interesting relation with integer partition problems. The problem 3-Partition is used for all NP-completeness reductions. The extension of interval graphs when the space in T ′ is limited is "equivalent" to the BinPacking problem.

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