Pavel Klavík

Extending partial representations of function graphs and permutation graphs

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear. We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbics comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0,1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0,1] becomes NP-complete.

Extending partial representations of interval graphs

We initiate the study of the computational complexity of the question of extending partial representations of geometric intersection graphs. In this paper we consider classes of interval graphs - given a collection of real intervals that forms an intersection representation of an induced subgraph of an input graph, is it possible to add intervals to achieve an intersection representation of the entire graph? We present an O(n2) time algorithm that solves this problem and constructs a representation if one exists. Our algorithm can also be used to list all nonisomorphic extensions with O(n2) delay. Although the classes of proper and unit interval graphs coincide, the partial representation extension problems differ on them. We present an O(mn) time decision algorithm for partial representation extension of proper interval graphs, but for unit interval graphs the complexity remains open. Finally we show how our methods can be used for solving the problem of simultaneous interval representations. We prove that this problem is fixed-paramater tractable with the size of the common intersection of the input graphs being the parameter.

Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs

A graph

Changing computing paradigms towards power efficiency

G

Extending Partial Representations of Subclasses of Chordal Graphs

covers a graph

Extending Partial Representations of Proper and Unit Interval Graphs

H

Automorphism Groups of Geometrically Represented Graphs

if there exists a locally bijective homomorphism from

Minimal Obstructions for Partial Representations of Interval Graphs

G

Structural and Complexity Aspects of Line Systems of Graphs

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3-connected reduction for regular graph covers

H