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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 $O((n+m)\alpha(n+m))$ where $n$ is the number of vertices, $m$ is the number of edges and $\alpha$ 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.

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 $\textrm{Aut}(G)$. 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, B\'ilka 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 $K_5$., Comment: The journal version of the second part of arXiv:1402.3774. arXiv admin note: substantial text overlap with arXiv:1402.3774

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.

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 $u \in V(G)$, we are given a list ${\mathfrak L}(u) \subseteq V(H)$ 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.

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 Gajarsk\'y 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 W2-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 Klav\'ik et al. show that it is polynomial-time solvable for $k$-nested interval graphs.

We investigate automorphism groups of planar graphs. The main result is a complete recursive description of all abstract groups that can be realized as automorphism groups of planar graphs. The characterization is formulated in terms of inhomogeneous wreath products. In the proof, we combine techniques from combinatorics, group theory, and geometry. This significantly improves the Babai's description (1975)., Comment: Some citations were added

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 ${\rm dim}(X)$ 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 ${\rm dim}(X) \le k$. 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 $k \ge 4$, 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.

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular coverings in which this homomorphism is prescribed by an action of a semiregular subgroup $\Gamma$ of $\textrm{Aut}(G)$; so $H \cong G / \Gamma$. In this paper, we study the behaviour of regular graph covering with respect to 1-cuts and 2-cuts in $G$. We describe reductions which produce a series of graphs $G = G_0,\dots,G_r$ such that $G_{i+1}$ is created from $G_i$ by replacing certain inclusion minimal subgraphs with colored edges. The process ends with a primitive graph $G_r$ which is either 3-connected, or a cycle, or $K_2$. This reduction can be viewed as a non-trivial modification of reductions of Mac Lane (1937), Trachtenbrot (1958), Tutte (1966), Hopcroft and Tarjan (1973), Cuningham and Edmonds (1980), Walsh (1982), and others. A novel feature of our approach is that in each step all essential information about symmetries of $G$ are preserved. A regular covering projection $G_0\to H_0$ induces regular covering projections $G_i \to H_i$ where $H_i$ is the $i$-th quotient reduction of $H_0$. This property allows to construct all possible quotients $H_0$ of $G_0$ from the possible quotients $H_r$ of $G_r$. By applying this method to planar graphs, we give a proof of Negami's Theorem (1988). Our structural results are also used in subsequent papers for regular covering testing when $G$ is a planar graph and for an inductive characterization of the automorphism groups of planar graphs (see Babai (1973) as well)., Comment: The journal version of the first part of arXiv:1402.3774

We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of interval, permutation and circle graphs. We combine techniques from group theory (products, homomorphisms, actions) with data structures from computer science (PQ-trees, split trees, modular trees) that encode all geometric representations. We prove that interval graphs have the same automorphism groups as trees, and for a given interval graph, we construct a tree with the same automorphism group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982]. For permutation and circle graphs, we give an inductive characterization by semidirect and wreath products. We also prove that every abstract group can be realized by the automorphism group of a comparability graph/poset of the dimension at most four.

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.

Power awareness is fast becoming immensely important in computing, ranging from the traditional High Performance Computing applications, to the new generation of data centric workloads. In this work we describe our efforts towards a power efficient computing paradigm that combines low precision and high precision arithmetic. We showcase our ideas for the widely used kernel of solving systems of linear equations that finds numerous applications in scientific and engineering disciplines as well as in large scale data analytics, statistics and machine learning. Towards this goal we developed tools for the seamless power profiling of applications at a fine grain level. In addition, we verify here previous work on post FLOPS/Watt metrics and show that these can shed much more light in the power/energy profile of important applications.

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science. We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for two 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. Therefore, we restrict ourselves to graph classes with polynomially solvable graph isomorphism. Inspired by Negami, we apply the structural results used by Babai in the 1970's to study automorphism groups of graphs. Our main result is the following FPT meta-algorithm: Let $\cal C$ be a class of graphs such that the structure of automorphism groups of 3-connected graphs in $\cal C$ is simple. Then we can solve RegularCover for $\cal C$-inputs $G$ in time $O^*(2^{e(H)/2})$ where $e(H)$ denotes the number of the edges of $H$. As one example of $\cal C$, this meta-algorithm applies to planar graphs. In comparison, testing general graph covers is known to be NP-complete for planar inputs $G$ even for small fixed graphs $H$ such as $K_4$ or $K_5$. Most of our results also apply to general graphs, in particular the complete structural understanding of regular covers for 2-cuts., Comment: The conference version accepted to ICALP 2014

The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph $G$ and a partial representation $\cal R'$ giving some pre-drawn chords that represent an induced subgraph of $G$. The question is whether one can extend $\cal R'$ to a representation $\cal R$ of the entire graph $G$, i.e., whether one can draw the remaining chords into a partially pre-drawn representation to obtain a representation of $G$. Our main result is an $O(n^3)$ time algorithm for partial representation extension of circle graphs, where $n$ is the number of vertices. To show this, we describe the structure of all representations of a circle graph using split decomposition. This can be of independent interest.

Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems.

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known 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 $\cal R'$ fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation $\cal R$ of the entire graph $G$ extending $\cal R'$. 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.

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 $\text{P} = \text{NP}$, 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.

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 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.

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 Golumbic's 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., Comment: Submitted to ESA 2012, track A

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 $P(S,G)$ 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 $\ell$-subgraph contractibility problem in which the edges of a set $S$ are required to form disjoint connected subgraphs of size at most $\ell$. This problem can be solved in time $O(n^2 f'(k,\ell))$ using the general algorithm. We also show that for $\ell \ge 2$ the problem is NP-complete.

The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar. PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvil asked whether there are non-trivial graphs for which Cover(H) is NP-complete but PlanarCover(H) belongs to P. We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of PlanarCover(H) in these cases., Comment: Full version (including Appendix) of a paper from the conference WG 2011