Pavel Klavík | Charles University, Prague (original) (raw)

Papers by Pavel Klavík

Research paper thumbnail of Cops and Robbers on Intersection Graphs

Algorithms and Computation, 2013

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on... more The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Research paper thumbnail of Isomorphisms of maps on the sphere

Contemporary mathematics, 2021

Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních g... more Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních grafů. V příspěvku navrhneme modifikovaný algoritmus na řešení tohoto problému s lineárnou složitostí. Práce obsahuje detaily nevyhnutné pro implementaci i důkaz linearity.For a class of objects with a well-defined isomorphism relation the isomorphism problem asks to determine the algorithmic complexity of the decision whether two given objects are, or are not, isomorphic. Theorems by Steinitz (1916), Whitney (1933) and Mani (1971) show that the isomorphism problems for convex polyhedra, for 3-connected planar graphs, and for the spherical maps are closely related. In 1974, Hopcroft and Wong investigated the complexity of the graph isomorphism problem for polyhedral graphs. They proved that the problem can be solved in linear time. We describe a modified linear-time algorithm solving the isomorphism problem for spherical maps based on the approach by Hopcroft and Wong. The paper includes a detailed description of the algorithm including proofs. Moreover, our modified algorithm allows to determine (in linear time) the group of orientation-preserving symmetries of a spherical map

Research paper thumbnail of Extending Partial Representations of Proper and Unit Interval Graphs

arXiv (Cornell University), Jul 30, 2012

The recently introduced problem of extending partial interval representations asks, for an interv... more 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.

Research paper thumbnail of Concept Map Assessment: An Experimental Mathematical Study

In this experimental study, we analyzed the ability to understand and ability to share mathematic... more In this experimental study, we analyzed the ability to understand and ability to share mathematical knowledge of our modified context maps (MCM) and compared them to the standard linear form of examination. For these purposes, the categorization of mathematical knowledge to local and structural understanding and craft was defined. Experimentation was conducted during the regular final oral exam of Linear algebra courses for computer science freshmen university students. No benefits were given for participation in the experiment. According to the questionnaire self-report student data, the MCM method combined with student-examiner discussion shares statistically significantly better structural understanding than the linear form. However, the MCM method shares less local understanding than the linear form, given randomized data set. Moreover, students claim that the MCM oral examination form is almost as objective as other oral exams they attempted during faculty study. Students created surprisingly good modified cognitive maps, although we assumed their low to none practical experience with concept mapping.

Research paper thumbnail of Extending Partial Representations of Subclasses of Chordal Graphs

arXiv (Cornell University), Jul 1, 2012

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of ... more 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.

Research paper thumbnail of On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

arXiv (Cornell University), Oct 14, 2015

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equ... more In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2. We propose generalizations of proper interval graphs called k-nested interval graphs in which there are no chains of k + 1 intervals nested in each other. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs. We give a linear-time recognition algorithm for k-nested interval graphs. This algorithm adds a missing piece to Gajarský et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting k and the length of the formula, while the problem is W[2]-hard when parameterized just by the length of the formula. We show that a generalization of recognition called partial representation extension is NP-hard for k-length interval graphs, even when k = 2, while Klavík et al. show that it is polynomial-time solvable for k-nested interval graphs.

Research paper thumbnail of Cops and Robbers on intersection graphs

European Journal of Combinatorics, Aug 1, 2018

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on... more The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Research paper thumbnail of MSOL Restricted Contractibility to Planar Graphs

arXiv (Cornell University), Apr 26, 2012

We study the computational complexity of graph planarization via edge contraction. The problem Co... more We study the computational complexity of graph planarization via edge contraction. The problem Contract asks whether there exists a set S of at most k edges that when contracted produces a planar graph. We work with a more general problem called P-RestrictedContract in which S, in addition, is required to satisfy a fixed MSOL formula P (S, G). We give an FPT algorithm in time O(n 2 f (k)) which solves P-RestrictedContract, where n is number of vertices of the graph and P (S, G) is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion-minimal solution S). As a specific example, we can solve the-subgraph contractibility problem in which the edges of the set S are required to form disjoint connected subgraphs of size at most. This problem can be solved in time O(n 2 f (k,)) using the general algorithm. We also show that for ≥ 2 the problem is NP-complete.

Research paper thumbnail of Extending Partial Representations of Interval Graphs

Lecture Notes in Computer Science, 2011

Interval graphs are intersection graphs of closed intervals of the real-line. The wellknown compu... more 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.

Research paper thumbnail of Automorphism Groups of Comparability Graphs

arXiv (Cornell University), Jun 16, 2015

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is t... more Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension dim(X) of a comparability graph X is the dimension of any transitive orientation of X, and by k-DIM we denote the class of comparability graphs X with dim(X) ≤ k. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordan's characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For k ≥ 4, we show that every finite group can be realized as the automorphism group of some graph in k-DIM, and testing graph isomorphism for k-DIM is GI-complete.

Research paper thumbnail of Circle Graph Isomorphism in Almost Linear Time

arXiv (Cornell University), Aug 24, 2019

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algo... more Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time O((n + m)α(n + m)) where n is the number of vertices, m is the number of edges and α is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time O(nm) of the previous algorithm [Hsu, 1995] based on a similar approach.

Research paper thumbnail of Graph isomorphism restricted by lists

Theoretical Computer Science, Mar 1, 2021

The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical comp... more The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs G and H, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each u ∈ V(G), we are given a list L(u) ⊆ V(H) of possible images of u. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates into NP-completeness of ListIso. 2) Combinatorial algorithms for GraphIso translate into algorithms for ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, bounded genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory do not translate: ListIso remains NP-complete for cubic colored graphs with sizes of color classes bounded by 8. Also, ListIso allows to classify results for the graph isomorphism problem. Some algorithms are robust and translate to ListIso. A fundamental problem is to construct a combinatorial polynomial-time algorithm for cubic graph isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP.

Research paper thumbnail of Algorithmic Aspects of Regular Graph Covers

arXiv (Cornell University), Sep 10, 2016

A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal ... more A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of Aut(G). We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for given graphs G and H whether G regularly covers H. When |H| = 1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem. Our main result is an involved FPT algorithm solving RegularCover for planar inputs G in time O * (2 e(H)/2) where e(H) denotes the number of edges of H. The algorithm is based on dynamic programming and employs theoretical results proved in a related structural paper. Further, when G is 3-connected, H is 2-connected or the ratio |G|/|H| is an odd integer, we can solve the problem RegularCover in polynomial time. In comparison, Bílka et al. (2011) proved that testing general graph covers is NP-complete for planar inputs G when H is a small fixed graph such as K 4 or K 5 .

Research paper thumbnail of Jordan-like characterization of automorphism groups of planar graphs

Journal of Combinatorial Theory, Series B

In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of ... more In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is then obtained as a semidirect product of a direct product of these stabilizers with a spherical group. The formulation of the main result is new and original. Moreover, it gives a deeper in the structure of the groups. As a consequence, automorphism groups of several subclasses of planar graphs, including 2-connected planar, outerplanar, and series-parallel graphs, are characterized. Our approach translates into a quadratic-time algorithm for computing the automorphism group of a planar graph which is the first such algorithm described in detail.

Research paper thumbnail of Isomorphisms of maps on the sphere

Polytopes and Discrete Geometry, 2021

Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních g... more Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních grafů. V příspěvku navrhneme modifikovaný algoritmus na řešení tohoto problému s lineárnou složitostí. Práce obsahuje detaily nevyhnutné pro implementaci i důkaz linearity.For a class of objects with a well-defined isomorphism relation the isomorphism problem asks to determine the algorithmic complexity of the decision whether two given objects are, or are not, isomorphic. Theorems by Steinitz (1916), Whitney (1933) and Mani (1971) show that the isomorphism problems for convex polyhedra, for 3-connected planar graphs, and for the spherical maps are closely related. In 1974, Hopcroft and Wong investigated the complexity of the graph isomorphism problem for polyhedral graphs. They proved that the problem can be solved in linear time. We describe a modified linear-time algorithm solving the isomorphism problem for spherical maps based on the approach by Hopcroft and Wong. The paper includes a detailed description of the algorithm including proofs. Moreover, our modified algorithm allows to determine (in linear time) the group of orientation-preserving symmetries of a spherical map

Research paper thumbnail of Circle Graph Isomorphism in Almost Linear Time

Lecture Notes in Computer Science, Aug 24, 2019

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algo... more Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time O((n + m)α(n + m)) where n is the number of vertices, m is the number of edges and α is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time O(nm) of the previous algorithm [Hsu, 1995] based on a similar approach.

Research paper thumbnail of Constructive Approach to Automorphism Groups of Planar Graphs

arXiv: Combinatorics, 2015

By Frucht's Theorem, every abstract finite group is isomorphic to the automorphism group of s... more By Frucht's Theorem, every abstract finite group is isomorphic to the automorphism group of some graph. In 1975, Babai characterized which of these abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is obtained as semidirect product of a direct product of these stabilizers with a spherical group. Our approach is based on the decomposition to 3-connected components and gives a quadratic-time algorithm for computing the automorphism group of a planar graph.

Research paper thumbnail of 3-CONNECTED Reduction for Regular Graph Covers

European Journal of Combinatorics, 2018

Research paper thumbnail of Structural and Complexity Aspects of Line Systems of Graphs

Lecture Notes in Computer Science, 2010

Research paper thumbnail of Extending Partial Representations of Proper and Unit Interval Graphs

Lecture Notes in Computer Science, 2014

The recently introduced problem of extending partial interval representations asks, for an interv... more 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.

Research paper thumbnail of Cops and Robbers on Intersection Graphs

Algorithms and Computation, 2013

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on... more The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Research paper thumbnail of Isomorphisms of maps on the sphere

Contemporary mathematics, 2021

Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních g... more Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních grafů. V příspěvku navrhneme modifikovaný algoritmus na řešení tohoto problému s lineárnou složitostí. Práce obsahuje detaily nevyhnutné pro implementaci i důkaz linearity.For a class of objects with a well-defined isomorphism relation the isomorphism problem asks to determine the algorithmic complexity of the decision whether two given objects are, or are not, isomorphic. Theorems by Steinitz (1916), Whitney (1933) and Mani (1971) show that the isomorphism problems for convex polyhedra, for 3-connected planar graphs, and for the spherical maps are closely related. In 1974, Hopcroft and Wong investigated the complexity of the graph isomorphism problem for polyhedral graphs. They proved that the problem can be solved in linear time. We describe a modified linear-time algorithm solving the isomorphism problem for spherical maps based on the approach by Hopcroft and Wong. The paper includes a detailed description of the algorithm including proofs. Moreover, our modified algorithm allows to determine (in linear time) the group of orientation-preserving symmetries of a spherical map

Research paper thumbnail of Extending Partial Representations of Proper and Unit Interval Graphs

arXiv (Cornell University), Jul 30, 2012

The recently introduced problem of extending partial interval representations asks, for an interv... more 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.

Research paper thumbnail of Concept Map Assessment: An Experimental Mathematical Study

In this experimental study, we analyzed the ability to understand and ability to share mathematic... more In this experimental study, we analyzed the ability to understand and ability to share mathematical knowledge of our modified context maps (MCM) and compared them to the standard linear form of examination. For these purposes, the categorization of mathematical knowledge to local and structural understanding and craft was defined. Experimentation was conducted during the regular final oral exam of Linear algebra courses for computer science freshmen university students. No benefits were given for participation in the experiment. According to the questionnaire self-report student data, the MCM method combined with student-examiner discussion shares statistically significantly better structural understanding than the linear form. However, the MCM method shares less local understanding than the linear form, given randomized data set. Moreover, students claim that the MCM oral examination form is almost as objective as other oral exams they attempted during faculty study. Students created surprisingly good modified cognitive maps, although we assumed their low to none practical experience with concept mapping.

Research paper thumbnail of Extending Partial Representations of Subclasses of Chordal Graphs

arXiv (Cornell University), Jul 1, 2012

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of ... more 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.

Research paper thumbnail of On the Classes of Interval Graphs of Limited Nesting and Count of Lengths

arXiv (Cornell University), Oct 14, 2015

In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equ... more In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called k-length interval graphs were considered in which the number of different lengths of intervals is limited by k. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for k = 2. We propose generalizations of proper interval graphs called k-nested interval graphs in which there are no chains of k + 1 intervals nested in each other. It is easy to see that k-nested interval graphs are a superclass of k-length interval graphs. We give a linear-time recognition algorithm for k-nested interval graphs. This algorithm adds a missing piece to Gajarský et al. [FOCS 2015] to show that testing FO properties on interval graphs is FPT with respect to the nesting k and the length of the formula, while the problem is W[2]-hard when parameterized just by the length of the formula. We show that a generalization of recognition called partial representation extension is NP-hard for k-length interval graphs, even when k = 2, while Klavík et al. show that it is polynomial-time solvable for k-nested interval graphs.

Research paper thumbnail of Cops and Robbers on intersection graphs

European Journal of Combinatorics, Aug 1, 2018

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on... more The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g + 15 in case of orientable surface of genus g, and at most 10g + 15 in case of non-orientable surface of Euler genus g. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, we show that the cop number is unbounded on intersection graphs of two-element subsets of a line, as well as on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Research paper thumbnail of MSOL Restricted Contractibility to Planar Graphs

arXiv (Cornell University), Apr 26, 2012

We study the computational complexity of graph planarization via edge contraction. The problem Co... more We study the computational complexity of graph planarization via edge contraction. The problem Contract asks whether there exists a set S of at most k edges that when contracted produces a planar graph. We work with a more general problem called P-RestrictedContract in which S, in addition, is required to satisfy a fixed MSOL formula P (S, G). We give an FPT algorithm in time O(n 2 f (k)) which solves P-RestrictedContract, where n is number of vertices of the graph and P (S, G) is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion-minimal solution S). As a specific example, we can solve the-subgraph contractibility problem in which the edges of the set S are required to form disjoint connected subgraphs of size at most. This problem can be solved in time O(n 2 f (k,)) using the general algorithm. We also show that for ≥ 2 the problem is NP-complete.

Research paper thumbnail of Extending Partial Representations of Interval Graphs

Lecture Notes in Computer Science, 2011

Interval graphs are intersection graphs of closed intervals of the real-line. The wellknown compu... more 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.

Research paper thumbnail of Automorphism Groups of Comparability Graphs

arXiv (Cornell University), Jun 16, 2015

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is t... more Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension dim(X) of a comparability graph X is the dimension of any transitive orientation of X, and by k-DIM we denote the class of comparability graphs X with dim(X) ≤ k. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordan's characterization for trees (1869). For permutation graphs, there is an extra operation, so there are some extra groups not realized by trees. For k ≥ 4, we show that every finite group can be realized as the automorphism group of some graph in k-DIM, and testing graph isomorphism for k-DIM is GI-complete.

Research paper thumbnail of Circle Graph Isomorphism in Almost Linear Time

arXiv (Cornell University), Aug 24, 2019

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algo... more Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time O((n + m)α(n + m)) where n is the number of vertices, m is the number of edges and α is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time O(nm) of the previous algorithm [Hsu, 1995] based on a similar approach.

Research paper thumbnail of Graph isomorphism restricted by lists

Theoretical Computer Science, Mar 1, 2021

The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical comp... more The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs G and H, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each u ∈ V(G), we are given a list L(u) ⊆ V(H) of possible images of u. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates into NP-completeness of ListIso. 2) Combinatorial algorithms for GraphIso translate into algorithms for ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, bounded genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory do not translate: ListIso remains NP-complete for cubic colored graphs with sizes of color classes bounded by 8. Also, ListIso allows to classify results for the graph isomorphism problem. Some algorithms are robust and translate to ListIso. A fundamental problem is to construct a combinatorial polynomial-time algorithm for cubic graph isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP.

Research paper thumbnail of Algorithmic Aspects of Regular Graph Covers

arXiv (Cornell University), Sep 10, 2016

A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal ... more A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of Aut(G). We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for given graphs G and H whether G regularly covers H. When |H| = 1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem. Our main result is an involved FPT algorithm solving RegularCover for planar inputs G in time O * (2 e(H)/2) where e(H) denotes the number of edges of H. The algorithm is based on dynamic programming and employs theoretical results proved in a related structural paper. Further, when G is 3-connected, H is 2-connected or the ratio |G|/|H| is an odd integer, we can solve the problem RegularCover in polynomial time. In comparison, Bílka et al. (2011) proved that testing general graph covers is NP-complete for planar inputs G when H is a small fixed graph such as K 4 or K 5 .

Research paper thumbnail of Jordan-like characterization of automorphism groups of planar graphs

Journal of Combinatorial Theory, Series B

In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of ... more In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is then obtained as a semidirect product of a direct product of these stabilizers with a spherical group. The formulation of the main result is new and original. Moreover, it gives a deeper in the structure of the groups. As a consequence, automorphism groups of several subclasses of planar graphs, including 2-connected planar, outerplanar, and series-parallel graphs, are characterized. Our approach translates into a quadratic-time algorithm for computing the automorphism group of a planar graph which is the first such algorithm described in detail.

Research paper thumbnail of Isomorphisms of maps on the sphere

Polytopes and Discrete Geometry, 2021

Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních g... more Hopcroft a Wong v roku 1974 navrhli lineárny algoritmus na zjišťování isomorfismu polyherálních grafů. V příspěvku navrhneme modifikovaný algoritmus na řešení tohoto problému s lineárnou složitostí. Práce obsahuje detaily nevyhnutné pro implementaci i důkaz linearity.For a class of objects with a well-defined isomorphism relation the isomorphism problem asks to determine the algorithmic complexity of the decision whether two given objects are, or are not, isomorphic. Theorems by Steinitz (1916), Whitney (1933) and Mani (1971) show that the isomorphism problems for convex polyhedra, for 3-connected planar graphs, and for the spherical maps are closely related. In 1974, Hopcroft and Wong investigated the complexity of the graph isomorphism problem for polyhedral graphs. They proved that the problem can be solved in linear time. We describe a modified linear-time algorithm solving the isomorphism problem for spherical maps based on the approach by Hopcroft and Wong. The paper includes a detailed description of the algorithm including proofs. Moreover, our modified algorithm allows to determine (in linear time) the group of orientation-preserving symmetries of a spherical map

Research paper thumbnail of Circle Graph Isomorphism in Almost Linear Time

Lecture Notes in Computer Science, Aug 24, 2019

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algo... more Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time O((n + m)α(n + m)) where n is the number of vertices, m is the number of edges and α is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time O(nm) of the previous algorithm [Hsu, 1995] based on a similar approach.

Research paper thumbnail of Constructive Approach to Automorphism Groups of Planar Graphs

arXiv: Combinatorics, 2015

By Frucht's Theorem, every abstract finite group is isomorphic to the automorphism group of s... more By Frucht's Theorem, every abstract finite group is isomorphic to the automorphism group of some graph. In 1975, Babai characterized which of these abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is obtained as semidirect product of a direct product of these stabilizers with a spherical group. Our approach is based on the decomposition to 3-connected components and gives a quadratic-time algorithm for computing the automorphism group of a planar graph.

Research paper thumbnail of 3-CONNECTED Reduction for Regular Graph Covers

European Journal of Combinatorics, 2018

Research paper thumbnail of Structural and Complexity Aspects of Line Systems of Graphs

Lecture Notes in Computer Science, 2010

Research paper thumbnail of Extending Partial Representations of Proper and Unit Interval Graphs

Lecture Notes in Computer Science, 2014

The recently introduced problem of extending partial interval representations asks, for an interv... more 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.