Generalized Nested Dissection

We initiate the study of the testability of properties in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time. The previous bound for this class of graphs was O(√n), and the constant-time testability was only known for planar graphs with bounded degree. Previously used transformations of unbounded-degree sparse graphs into bounded- degree sparse graphs cannot be used to reduce the problem to the testability of bounded-degree planar graphs. Our approach extends to arbitrary minor-free graphs. Our algorithm is based on random walks. The challenge here is to analyze random walks for a class of graphs that has good separators, i.e., bad expansion. Standard techniques that use a fast convergence to a uniform distribution do not work in this case. Roughly speaking, our analysis technique self-reduces the problem of finding an odd-length cycle in a multigraph G induced by a collection of cycles to another multigraph G' induced by a set of shorter odd-length cycles, in such a way that when a random walks finds a cycle in G' with probability p >; 0, then it does so with probability λ(p) >; 0 in G. This reduction is applied until the cycles collapse to self-loops that can be easily detected.

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.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

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.

On Graphs Which Contain All 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-first search is among the hardest problems on planar 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.

Improved induced matchings in sparse graphs

Efficient approximate solution of sparse linear systems

The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs where vertices of the two parts lie on two horizontal lines and edges lie between these lines. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been considered only for $k=1$ in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of $2$-layer $k$-planar graphs with $k\in \{2,3,4,5\}$. Based on these results, we provide a Crossing Lemma for $2$-layer $k$-planar graphs, which then implies a general density bound for $2$-layer $k$-planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between $k$-planarity and $h$-quasiplanarity in the $2$-layer model and show that $2$-layer $k$-planar graphs have pathwidth at most $k+1$ while there are also $2$-layer $k$-planar graphs with pathwidth at least $(k+3)/2$.

Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant e >0, there is a polynomial-time 1+e-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+e for any e>0 in polynomial time. However, the degree of the polynomial grows with 1∕e and hence, the approximation scheme is hardly practical.

Edge Separators of Planar and Outerplanar Graphs With Applications

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $O(p^3\log p)$ colors where the previous bound was $O(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $O(p)$ colors while it was conjectured that they may require exponential number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring. This bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth $3$. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

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.