Geometric Representations of Graphs
A prime example of geometrically represented graphs is the class of planar graphs. Another examples are
classes of intersection graphs in which each vertex is represented by some set and two vertices are adjacent if
and only if the corresponding sets intersect. For instance, interval graphs are intersection graphs
of closed intervals of the real line.
An interval graph with two different interval
representations, with orderings of maximal cliques depicted.
For more information and detailed description of our results, see my PhD thesis
Extension Properties of Graphs and Structures.
Structure of all representations
Nicely behaving classes of geometrically represented graphs have the property that all possible
geometric representations can be efficiently described and understood. For instance, interval graphs
have MPQtrees of Booth and Lueker (1979), Korte and Möhring (1989) describing all possible interval
representations by one tree data structure.
Two equivalent MPQtrees corresponding to two above interval
representations.
For planar graphs, decomposition into 3connected components (which have unique embeddings into
the sphere) was pioneered by Mac Lane (1937) and Trakhtenbrot (1958). It was followed by multiple
papers and it was recently popularized under the name SPQR trees. In [1], we have described an
augmented version of 3connected decomposition which captures automorphism groups. For description
of our results and comparison with previous results, see Chapters 6
and 7 of my PhD thesis.
This is related to the research project Symmetries and graph
isomorphism.
On the left, a graph G. On the right, the 3connected
decomposition of G captured by a tree.
The partial representation extension problems
This problem is already from my bachelor's thesis from 2010. Given a graph and a partial
intersection representation, we ask whether this partial representation can be extended to a full
representation of the input. For instance for interval graphs, some intervals are predrawn, fixed
by the input, and we ask whether the remaining interval can be added to form the full interval
representation.
Comparison of the recognition problems (Recog) and of the
partial representation extension problems (RepExt).
For interval graphs, we prove the following results:
Theorem: [2,4]
There exists a lineartime certifying algorithm for the partial representation extension problem of
interval graphs. If the answer is "yes", it outputs an extending representation. If the answer is
"no", it detects one of the minimal obstructions.
Concerning structural characterizations, we generalize both FulkersonGross and LekkerkerkerBoland
to partially represented interval graphs. In [2], we prove that a partial interval representation is
extendible if and only if there exists a consecutive ordering of maximal cliques extending some
partial ordering derived from the partial representation. In [4], we characterize minimal
obstructions which make a partial interval representation nonextendible. See Chapters 2, 3, and 4
of
my PhD thesis.
Currently, the complexity of the partial representation extension problems is understood for a
variety of classes of intersection graphs. In [3], we describe a lineartime algorithm for proper
interval graphs and almost quadratictime algorithm for unit interval graphs. In [5], we describe a
cubictime algorithm for circle graphs.
Hasse diagram of the main classes of geometrically
represented graphs together with the complexity of the partial representation extension problems.
For references, see page 57 of my PhD thesis.
Cops and robber on intersection graphs
In [6], we have studied the game of Cops and Robber on intersection graphs. We prove the following
results:
Theorem: [6]
For every connected string graph, 15 cops are sufficient to catch the robber.
On the other hand, for many basic classes of intersection graphs of disconnected or spacial sets, we
prove that the needed number of cops in unbounded.
Selected publications

Jiří Fiala, Pavel Klavík, Jan Kratochvíl, and Roman Nedela:
Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs.
In Automata, Languages, and Programming, ICALP 2014, volume 8572 of Lecture Notes in Computer Science, pages 489501, 2014.
Links: arXiv, proceedings link.

Pavel Klavík, Jan Kratochvíl, Yota Otachi, Toshiki Saitoh, and Tomáš Vyskočil:
Extending Partial Representations of Interval Graphs.
Algorithmica 78(3):945967, 2017.
The conference version appeared in Theory and Applications of Models of Computation, TAMC 2011,
volume 6648 of Lecture Notes in Computer Science, pages 276285, 2011.
Links: arXiv, journal link.

Pavel Klavík, Jan Kratochvíl, Yota Otachi, Ignaz Rutter, Toshiki Saitoh, Maria Saumell, and Tomáš Vyskočil:
Extending Partial Representations of Proper and Unit Interval Graphs.
Algorithmica 77(4):10711104, 2017.
The conference version appeared in Algorithm Theory, SWAT 2014, volume 8503 of Lecture Notes in Computer Science, pages 253264, 2014.
Links: arXiv, journal link.

Pavel Klavík, and Maria Saumell:
Minimal Obstructions for Partial Representations of Interval Graphs.
Submitted, 2015.
The conference version appeared in Algorithms and Computation, ISAAC 2014, volume 8889 of Lecture Notes in Computer Science, pages 401413, 2014.
Links: arXiv, proceedings link.

Steve Chaplick, Radoslav Fulek, and Pavel Klavík:
Extending Partial Representations of Circle Graphs.
Submitted, 2015.
The conference version appeared in Graph Drawing, GD 2013, volume 8242 of Lecture Notes in Computer Science, pages 131142, 2013.
Links: arXiv, proceedings link.

Tomáš Gavenčiak, Przemysław Gordinowicz, Vít Jelínek, Pavel Klavík, and Jan Kratochvíl:
Cops and Robbers on Intersection Graphs.
Submitted, 2016.
The conference versions appeared in Algorithms and Computation, ISAAC 2013, volume 8283 of
Lecture Notes in Computer Science, pages 174184, 2013,
and in Algorithms and Computation, ISAAC 2015, volume 9472 of Lecture Notes in Computer Science, pages 355366, 2015.
Links: arXiv,
proceedings link ISAAC 2013,
proceedings link ISAAC 2015.