On Graphs Which Contain All Sparse Graphs

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\text{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.

J. A. George has discovered a method, called nested dissection, for solving a system of linear equations defined on an n = k $\times$ k square grid in O(n log n) space and O($n{3/2}$) time. We generalize this method without degrading the time and space bounds so that it applies to any system of equations defined on a planar or almost-planar graph. Such systems arise in the solution of two-dimensional finite element problems. Our method uses the fact that planar graphs have good separators. More generally, we show that sparse Gaussian elimination is efficient for any class of graphs which have good separators, and conversely that graphs without good separators (including almost all sparse graphs) are not amenable to sparse Gaussian elimination.

Edge-disjoint odd cycles in planar graphs

Efficient approximate solution of sparse linear systems

Spanners in sparse graphs

The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O (log * n )-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-first search is among the hardest problems on planar graphs.

Improved induced matchings in sparse graphs

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-firstsearch is among the hardest problems on planar graphs.KeywordsPlanar GraphExternal MemorySimple CycleOuterplanar GraphBoundary CycleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This thesis studies exponential time algorithms that give optimum solutions to optimization problems which are highly unlikely to be solvable to optimality in polynomial time. Such algorithms are designed and upper bounds on their running times are analyzed, for pro-blems from graph theory, constraint satisfaction, and computational biology. It is studied how restrictions on one computationally hard parameter affect the running time of algorithms for computing another parameter. Algorithms are obtained to compute minimum dominating sets, minimum bandwidth and topological bandwidth layouts, and other hard parameters, in time exponential only in the maximum number of leaves in a spanning tree of a graph. For sparse graphs, algorithms find minimum connected dominating sets in time exponential only in the optimum value. These algorithms are based on kernels of small size, meaning that arbitrary instances are polynomial-time compressible to equivalent instances whose size depends on the parameter alone. Lower and upper bounds on the number and size of certain combinatorial objects are proved. For minimal feedback vertex sets in tournaments, improved bounds on their number and a polynomial-space polynomial-delay algorithm for their enumeration are given. For subcubic planar graphs, fast algorithms finding large induced matchings are given. Fixed-parameter algorithms are presented for maximization problems whose optimum solution value grows as an unbounded function with the instance size. Polynomial kernels are obtained for all permutation constraint satisfaction problems with ternary constraints, and independent set problems in restricted graph classes, when these problems are parameterized above tight lower bounds on their solution value. Polynomial space algorithms with subexponential time requirement are obtained for the minimum triplet inconsistency problem in phylogenetics, and basically all bidimensional problems in sparse graphs such as graphs excluding a fixed minor.

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

We describe efficient deterministic techniques for breaking symmetry in parallel. The techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in Ο(lg*n) time on an EREW PRAM using a linear number of processors. We apply these techniques to construct fast linear processor algorithms for several problems, including (Δ + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees in planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

Homomorphisms from sparse graphs with large girth

For a graph G, a set D⊆V(G) is called a [1,j]-dominating set if every vertex in V(G)∖D has at least one and at most j neighbors in D. A set D⊆V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study the aforementioned problems on various graph classes from the perspective of parameterized complexity and prove the following results:•[1,j]-Dominating Set parameterized by solution size is W[1]-hard on d-degenerate graphs for d=j+1.•[1,j]-Dominating Set parameterized by solution size is FPT on nowhere dense graphs.•The known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH).•Assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.•Finally, we study another variant of Dominating Set, called Restrained Dominating Set, that asks if there is a dominating set D of G of size at most k such that no vertex outside of D has all of its neighbors in D. We prove this variant is W[1]-hard even on 3-degenerate graphs.

A vertex-subset graph problem \(\mathcal {Q}\) defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions \(S_s\) and \(S_t\) of size k, whether it is possible to transform \(S_s\) into \(S_t\) by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by \(\textsc {ISR}\) and the latter by \(\textsc {DSR}\). Both \(\textsc {ISR}\) and \(\textsc {DSR}\) are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that \(\textsc {ISR}\) is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that \(\textsc {ISR}\) is fixed-parameter tractable on planar graphs and graphs of bounded degree. For \(\textsc {DSR}\), we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes degenerate and nowhere dense graphs.