Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast

The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph H of a given graph G such that H has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski\u27s famous planarity criterion. We build upon this approach and present new constraint classes - together with a lifting of the polyhedron - to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler\u27s polyhedron formula as a starting point and combine it with considering cycles in G. This paper discusses both the theoretical as well as the practical sides of this strengthening

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity

Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity

Planar Disjoint Paths, Treewidth, and Kernels

In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to find $k$ pairwise vertex-disjoint paths such that the $i$-th path connects $s_i$ to $t_i$. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by $k$ unless coNP $\subseteq$ NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by $k + tw$, where $tw$ denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to $tw= k^{O(1)}$ under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time $2^{O(k^2)}\cdot n^{O(1)}$, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

MacLane's Theorem for Graph-Like Spaces

The cycle space of a finite graph is the subspace of the edge space generated by the edge sets of cycles, and is a well-studied object in graph theory. Recently progress has been made towards extending the theory of cycle spaces to infinite graphs. Graph-like spaces are a class of topological objects that reconcile the combinatorial properties of infinite graphs with the topological properties of finite graphs. They were first introduced by Thomassen and Vella as a natural, general class of topological spaces for which Menger's Theorem holds. Graph-like spaces are the natural objects for extending classical results from topological graph theory and cycle space theory to infinite graphs. This thesis focuses on the topological properties of embeddings of graph-like spaces, as well as the algebraic properties of graph-like spaces. We develop a theory of embeddings of graph-like spaces in surfaces. We also show how the theory of edge spaces developed by Vella and Richter applies to graph-like spaces. We combine the topological and algebraic properties of embeddings of graph-like spaces in order to prove an extension of MacLane's Theorem. We also extend Thomassen's version of Kuratowski's Theorem for 2-connected compact locally connected metric spaces to the class of graph-like spaces