Abstract: "We prove that every n-vertex graph of genus g and maximal degree k has an edge separator of size O([square root of gkn]). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers."

Graph Separators with Applications is devoted to techniques for obtaining upper and lower bounds on the sizes of graph separators - upper bounds being obtained via decomposition algorithms. The book surveys the main approaches to obtaining good graph separations, while the main focus of the book is on techniques for deriving lower bounds on the sizes of graph separators. This asymmetry in focus reflects our perception that the work on upper bounds, or algorithms, for graph separation is much better represented in the standard theory literature than is the work on lower bounds, which we perceive as being much more scattered throughout the literature on application areas. Given the multitude of notions of graph separator that have been developed and studied over the past (roughly) three decades, there is a need for a central, theory-oriented repository for the mass of results. The need is absolutely critical in the area of lower-bound techniques for graph separators, since these techniques have virtually never appeared in articles having the word `separator' or any of its near-synonyms in the title. Graph Separators with Applications fills this need.

Many divide-and-conquer algorithms on graphs are based on finding a small set of vertices or edges whose removal divides the graph roughly in half. Most graphs do not have the necessary small separators, but some useful classes do. One such class is planar graphs: If we can draw an n-vertex graph on the plane, then we can bisect it by removing $O(\sqrt{n})$ vertices [Lipt79b].

This book constitutes the refereed proceedings of the 15th Algorithms and Data Structures Symposium, WADS 2017, held in St. John's, NL, Canada, in July/August 2017. The 49 full papers presented together with 3 abstracts of invited talks were carefully reviewed and selected from 109 submissions. They present original research on the theory and application of algorithms and data structures in many areas, including combinatorics, computational geometry, databases, graphics, and parallel and distributed computing. The WADS Symposium, which alternates with the Scandinavian Symposium and Workshops on Algorithm Theory, SWAT, is intended as a forum for researchers in the area of design and analysis of algorithms and data structures. Papers presenting original research on the theory and application of algorithms and data structures

This book contains Volumes 4 and 5 of the Journal of Graph Algorithms and Applications (JGAA) . The first book of this series, Graph Algorithms and Applications 1, published in March 2002, contains Volumes 1OCo3 of JGAA . JGAA is a peer-reviewed scientific journal devoted to the publication of high-quality research papers on the analysis, design, implementation, and applications of graph algorithms. Areas of interest include computational biology, computational geometry, computer graphics, computer-aided design, computer and interconnection networks, constraint systems, databases, graph drawing, graph embedding and layout, knowledge representation, multimedia, software engineering, telecommunications networks, user interfaces and visualization, and VLSI circuit design. The journal is supported by distinguished advisory and editorial boards, has high scientific standards, and takes advantage of current electronic document technology. The electronic version of JGAA is available on the Web at http: //jgaa.info/. Graph Algorithms and Applications 2 presents contributions from prominent authors and includes selected papers from the Dagstuhl Seminar on Graph Algorithms and Applications and the Symposium on Graph Drawing in 1998. All papers in the book have extensive diagrams and offer a unique treatment of graph algorithms focusing on the important applications. Contents: Approximations of Weighted Independent Set and Hereditary Subset Problems (M M Halldrsson); Approximation Algorithms for Some Graph Partitioning Problems (G He et al.); Geometric Thickness of Complete Graphs (M B Dillencourt et al.); Techniques for the Refinement of Orthogonal Graph Drawings (J M Six et al.); Navigating Clustered Graphs Using Force-Directed Methods (P Eades & M L Huang); Clustering in Trees: Optimizing Cluster Sizes and Number of Subtrees (S E Hambrusch et al.); Planarizing Graphs OCo A Survey and Annotated Bibliography (A Liebers); Fully Dynamic 3-Dimensional Orthogonal Graph Drawing (M Closson et al.); 1-Bend 3-D Orthogonal Box-Drawings: Two Open Problems Solved (T Biedl); Computing an Optimal Orientation of a Balanced Decomposition Tree for Linear Arrangement Problems (R Bar-Yehuda et al.); New Bounds for Oblivious Mesh Routing (K Iwama et al.); Connectivity of Planar Graphs (H de Fraysseix & P O de Mendez); and other papers. Readership: Researchers and practitioners in theoretical computer science, computer engineering, and combinatorics and graph theory."