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.

Gaussian elimination, which has been shown to be applicable to the solution problems in many different domains, is the technique used by COSMOS to symbolically analyze digital MOS networks for their behavior in terms of Boolean expressions. While pivot selection algorithms are known which minimize the total number of operations required to solve a system, this report will focus on pivot selection algorithms that result in expressions of small depth, from which fine-grained parallelism may be extracted. A graph theoretic approach to Gaussian elimination is adopted which allows the structure of sparse systems to be clearly examined, and an elimination ordering based on graph separators is shown to result in expressions of small depth. This report proposes an algorithm related to Gaussian elimination which characterizes graphs in terms of decomposition rules and shows that for graphs which may be reduced by an elimination ordering the results in low total complexity, a reordered elimination sequence may result in expressions of small depth.

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."

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

Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only O(square root of n) vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results. (Author).