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1. A Structure Theorem for Pseudo-Segments and Its Applications

   https://doi.org/10.4230/LIPIcs.SoCG.2024.59  

   Fox, Jacob; Pach, János; Suk, Andrew (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges. 

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2. Universal Geometric Graphs

   https://doi.org/10.1007/978-3-030-60440-0_14  

   Frati, Fabrizio; Hoffmann, Michael; Toth, Csaba D. (October 2020, Graph-Theoretic Concepts in Computer Science (WG 2020))

   null (Ed.)

   We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is universal for a class H of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in H . Our main result is that there exists a geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an n-vertex graph with O(nlogn) edges that contains every n-vertex forest as a subgraph. Our O(nlogn) bound on the number of edges is asymptotically optimal. We also prove that, for every h>0 , every n-vertex convex geometric graph that is universal for the class of the n-vertex outerplanar graphs has Ωh(n2−1/h) edges; this almost matches the trivial O(n2) upper bound given by the n-vertex complete convex geometric graph. Finally, we prove that there is an n-vertex convex geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex caterpillars. 

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3. How to morph graphs on the torus

   Chambers, Erin Wolf (January 2021, Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in O(n1+ω/2) time, where ω is the matrix multiplication exponent, and the computed morph consists of O(n) parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs. 

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4. Many Cliques with Few Edges

   https://doi.org/10.37236/9550  

   Kirsch, Rachel; Radcliffe, A. J. (January 2021, The Electronic Journal of Combinatorics)

   null (Ed.)

   Recently Cutler and Radcliffe proved that the graph on $$n$$ vertices with maximum degree at most $$r$$ having the most cliques is a disjoint union of $$\lfloor n/(r+1)\rfloor$$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $$m$$ edges and maximum degree at most $$r$$, the graph that has the most cliques of size at least two is the disjoint union of $$\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$$ cliques of size $r+1$ together with the colex graph using the remainder of the edges. 

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5. Stochastic Minimum Vertex Cover in General Graphs: A 3/2-Approximation

   https://doi.org/10.1145/3564246.3585230  

   Derakhshan, Mahsa; Durvasula, Naveen; Haghtalab, Nika (June 2023, ACM)

   We study the stochastic vertex cover problem. In this problem, G = (V, E) is an arbitrary known graph, and G⋆ is an unknown random subgraph of G where each edge e is realized independently with probability p. Edges of G⋆ can only be verified using edge queries. The goal in this problem is to find a minimum vertex cover of G⋆ using a small number of queries. Our main result is designing an algorithm that returns a vertex cover of G⋆ with size at most (3/2+є) times the expected size of the minimum vertex cover, using only O(n/є p) non-adaptive queries. This improves over the best-known 2-approximation algorithm by Behnezhad, Blum and Derakhshan [SODA’22] who also show that Ω(n/p) queries are necessary to achieve any constant approximation. Our guarantees also extend to instances where edge realizations are not fully independent. We complement this upperbound with a tight 3/2-approximation lower bound for stochastic graphs whose edges realizations demonstrate mild correlations. 

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