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1. Polygons with Prescribed Angles in 2D and 3D

   https://doi.org/10.1007/978-3-030-68766-3_11  

   Efrat, Alon; Fulek, Radoslav; Kobourov, Stephen; Toth, Csaba D. (February 2021, Graph Drawing and Network Visualization)

   null (Ed.)

   We consider the construction of a polygon P with n vertices whose turning angles at the vertices are given by a sequence A=(α0,…,αn−1) , αi∈(−π,π) , for i∈{0,…,n−1} . The problem of realizing A by a polygon can be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an angle graph. In 2D, we characterize sequences A for which every generic polygon P⊂R2 realizing A has at least c crossings, for every c∈N , and describe an efficient algorithm that constructs, for a given sequence A, a generic polygon P⊂R2 that realizes A with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence A can be realized by a (not necessarily generic) polygon P⊂R3 , and for every realizable sequence the algorithm finds a realization. 

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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. Packing Trees into 1-Planar Graphs

   https://doi.org/10.1007/978-3-030-39881-1_8  

   De Luca, F.; Di Giacomo, E.; Hong, S.; Kobourov, S.; Lenhart, W.; Liotta, G.; Meijer, H.; Tappini, A.; Wismath, S. (January 2020, 14th International Conference and Workshop on Algorithms and Computation (WALCOM))

   We introduce and study the 1-planar packing problem: Given k graphs with n vertices 𝐺1,…,𝐺𝑘, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each 𝐺𝑖 is a tree and 𝑘=3 . We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with 𝑛≥12 vertices admits a 1-planar packing, while such a packing does not exist if 𝑛≤10 . 

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4. Computing β-Stretch Paths in Drawings of Graphs

   Arkin, E; Darabi, F; Efrat, A; Frank, F; Fulek, R; Kobourov, S; Mitchell, J (October 2020, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT))

   Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P , the length of the sub-curve of π connecting f(p) with f(q) is at most β∥f(p) − f(q)∥, where ∥.∥ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. We also output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log |V (G)|)), and s, t ∈ V (G), outputs a β-stretch path between s and t, if a (1 − ε)β-stretch path between s and t exists in the drawing. 

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5. Random Graph Matching with Improved Noise Robustness

   Mao, Cheng; Rudelson, Mark; Tikhomirov, Konstantin (August 2021, Proceedings of Machine Learning Research)

   Belkin, Mikhail; Kpotufe, Samory (Ed.)

   Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. Recently, there has been a plethora of work studying efficient algorithms for graph matching under probabilistic models. In this work, we propose a new algorithm for graph matching: Our algorithm associates each vertex with a signature vector using a multistage procedure and then matches a pair of vertices from the two graphs if their signature vectors are close to each other. We show that, for two Erdős–Rényi graphs with edge correlation $$1-\alpha$$, our algorithm recovers the underlying matching exactly with high probability when $$\alpha \le 1 / (\log \log n)^C$$, where $$n$$ is the number of vertices in each graph and $$C$$ denotes a positive universal constant. This improves the condition $$\alpha \le 1 / (\log n)^C$$ achieved in previous work. 

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