Boundary classes for graph problems involving non-local properties

We continue the study of boundary classes for NP-hard problems and focus on seven NP-hard graph problems involving non-local properties: HAMILTONIAN CYCLE, HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE, HAMILTONIAN PATH, FEEDBACK VERTEX SET, CONNECTED VERTEX COVER, CONNECTED DOMINATING SET and GRAPH VCCON DIMENSION. Our main result is the determination of the first boundary class for FEEDBACK VERTEX SET. We also determine boundary classes for HAMILTONIAN CYCLE THROUGH SPECIFIED EDGE and HAMILTONIAN PATH and give some insights on the structure of some boundary classes for the remaining problems

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

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Graph Theory

This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results