On retracts, absolute retracts, and folds in cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

FO Model Checking of Geometric Graphs

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures - culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarsk\'y et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs $G_1$ and $G_2$, one looks for a graph with the maximum number of vertices being both an induced subgraph of $G_1$ and $G_2$. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of \textsc{Clique}, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by $k := \text{vc}(G_1)+\text{vc}(G_2)$ the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time $2^{O(k \log k)}$. We complement this result by showing that, unless the ETH fails, it cannot be solved in time $2^{o(k \log k)}$. This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by $k$, unless $NP \subseteq \mathsf{coNP}/poly$. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.Comment: This version introduces new result

Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs *

Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper interval graphs and on bipartite permutation graphs. We show that ISI can be solved in polynomial time on proper interval graphs and on bipartite permutation graphs, provided that H is connected. As a consequence, we obtain that ISI is fixed-parameter tractable on these two graph classes, when parametrised by the number of connected components of H. Our results contrast and complement the following known results: W [1]-hardness of ISI on interval graphs when parametrised by the number of vertices of H, NP-completeness of ISI on connected interval graphs and on connected permutation graphs, and NP-completeness of Subgraph Isomorphism on connected proper interval graphs and connected bipartite permutation graphs

Interactive Event Sequence Query and Transformation

In our burgeoning world of pervasive sensors and affordable data storage, records of timestamped events are being produced across nearly every domain of personal and professional computing. This temporal event data is a fundamental component of electronic health records, process logs, sports analytics, and more. Across all domains, however, are two overarching needs: (1) to understand population-level trends and patterns, and (2) to identify important subsets of individual records. Visual analytics tools are billed as the solution to both of these problems. A huge volume of work has demonstrated the ability of these tools to facilitate user-guided data exploration and hypothesis generation across a wide range of data types. What is typically ignored however, is the process that takes place between the data collection and this exploration stage, a process frequently referred to as data wrangling. For many data types, wrangling consists mostly of restructuring spreadsheet columns and renaming fields. For temporal event data though, this wrangling process can extend much further---to the data itself---where event patterns must be transformed to better reflect either the real world events that generated them or the perspective of a given study. Without this step, population-level trends can be obscured beyond the point of recognition, and important subsets of records are impossible to discern. Temporal event data wrangling, however, is deceivingly difficult and error prone even for expert users. Standard, command-based query languages are poorly suited for specifying even the simplest event patterns and, in systems that are not precisely designed for handling temporal constructs, these queries are executed using a series of slow and inefficient self-join operations. Attempts at more accessible query languages frequently omit critical features such as events that occur over a period of time (intervals) or the absence of an event. Perhaps most importantly is that query alone is not enough to get users through a typical temporal event data wrangling process. Event patterns not only need to be found, but also transformed and re-represented. Temporal event wrangling is just as much about revisal as it is about retrieval, and given the ubiquity of this data type, an effective solution on this front has the potential to hugely impact the way that we utilize this data to inform future decisions. An improved query and wrangling process would not only benefit database professionals, but also dramatically increase the range of users who can access this type of data, particularly domain expert medical researchers. This dissertation demonstrates the ability of the EventFlow visualization tool to extend beyond the typical bounds of data exploration, and serve as a critical aid for both temporal event query and data transformation. I begin by establishing a better understanding of why these two processes are innately error prone, and introduce a simple set of powerful yet usable mechanisms that can help reduce an initial portion of these errors. I then show that by coupling these mechanisms with interactive visualizations, users are able to both identify remaining errors and leverage those errors to construct more accurate queries and transformations. The direct contributions of this dissertation are (1) a graphic-based query capabilities over points, intervals, and absences, (2) an integer programming strategy for processing temporal queries, (3) a Find & Replace system for transforming event sequences, and (4) eight case studies that demonstrate the utility and validity of these approaches. However, this work is designed more broadly to open new avenues of research in how visualization and visual analytics tools can be leveraged for tasks beyond data exploration