The Graph Motif problem parameterized by the structure of the input graph

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research

Cluster Editing: Kernelization based on Edge Cuts

Kernelization algorithms for the {\sc cluster editing} problem have been a popular topic in the recent research in parameterized computation. Thus far most kernelization algorithms for this problem are based on the concept of {\it critical cliques}. In this paper, we present new observations and new techniques for the study of kernelization algorithms for the {\sc cluster editing} problem. Our techniques are based on the study of the relationship between {\sc cluster editing} and graph edge-cuts. As an application, we present an ${\cal O}(n^2)$-time algorithm that constructs a $2k$ kernel for the {\it weighted} version of the {\sc cluster editing} problem. Our result meets the best kernel size for the unweighted version for the {\sc cluster editing} problem, and significantly improves the previous best kernel of quadratic size for the weighted version of the problem

Calliope-Net: Automatic Generation of Graph Data Facts via Annotated Node-link Diagrams

Graph or network data are widely studied in both data mining and visualization communities to review the relationship among different entities and groups. The data facts derived from graph visual analysis are important to help understand the social structures of complex data, especially for data journalism. However, it is challenging for data journalists to discover graph data facts and manually organize correlated facts around a meaningful topic due to the complexity of graph data and the difficulty to interpret graph narratives. Therefore, we present an automatic graph facts generation system, Calliope-Net, which consists of a fact discovery module, a fact organization module, and a visualization module. It creates annotated node-link diagrams with facts automatically discovered and organized from network data. A novel layout algorithm is designed to present meaningful and visually appealing annotated graphs. We evaluate the proposed system with two case studies and an in-lab user study. The results show that Calliope-Net can benefit users in discovering and understanding graph data facts with visually pleasing annotated visualizations

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

A Generalization of Nemhauser and Trotter\u27s Local Optimization Theorem

The Nemhauser-Trotter local optimization theorem applies to the NP-hard textsc{Vertex Cover} problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter\u27s result to vertex deletion and graph packing problems, introducing novel algorithmic strategies based on purely combinatorial arguments (not referring to linear programming as the Nemhauser-Trotter result originally did). We exhibit our framework using a generalization of textsc{Vertex Cover}, called textrm{sc Bounded-Degree Deletion}, that has promise to become an important tool in the analysis of gene and other biological networks. For some fixed~$dgeq 0$, textrm{sc Bounded-Degree Deletion} asks to delete as few vertices as possible from a graph in order to transform it into a graph with maximum vertex degree at most~$d$. textsc{Vertex Cover} is the special case of $d=0$. Our generalization of the Nemhauser-Trotter theorem implies that textrm{sc Bounded-Degree Deletion} has a problem kernel with a linear number of vertices for every constant~$d$. We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves. Finally, charting the border between (parameterized) tractability and intractability for textrm{sc Bounded-Degree Deletion}, we provide a W[2]-hardness result for textrm{sc Bounded-Degree Deletion} in case of unbounded $d$-values