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1. 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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2. ENERGY COMPACTION FILTERS ON GRAPHS

   https://doi.org/10.1109/GlobalSIP.2018.8646570  

   Teke, Oguzhan ; Vaidyanathan, P. P. ( November 2018 , IEEE Global Conference on Signal and Information Processing)

   In classical signal processing spectral concentration is an important problem that was first formulated and analyzed by Slepian. The solution to this problem gives the optimal FIR filter that can confine the largest amount of energy in a specific bandwidth for a given filter order. The solution is also known as the prolate sequence. This study investigates the same problem for polynomial graph filters. The problem is formulated in both graph-free and graph-dependent fashions. The graph-free formulation assumes a continuous graph spectrum, in which case it becomes the polynomial concentration problem. This formulation has a universal approach that provides a theoretical reference point. However, in reality graphs have discrete spectrum. The graph-dependent formulation assumes that the eigenvalues of the graph are known and formulates the energy compaction problem accordingly. When the eigenvalues of the graph have a uniform distribution, the graph-dependent formulation is shown to be asymptotically equivalent to the graph-free formulation. However, in reality eigenvalues of a graph tend to have different densities across the spectrum. Thus, the optimal filter depends on the underlying graph operator, and a filter cannot be universally optimal for every graph. 

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3. Unavoidable Patterns in Complete Simple Topological Graphs

   Andrew Suk, Ji Zeng ( January 2023 , Graph Drawing and Network Visualization: 30th International Symposium, GD 2022, Tokyo, Japan, September 13–16, 2022, Revised Selected Papers)

   Angelini, P. (Ed.)

   We show that every complete n-vertex simple topological graph contains a topological subgraph on at least (logn)1/4−o(1) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound Ω(log1/8n) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least (logn)1−o(1) . 

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4. Effects of Tightening Unit-level and System-level Constraints in Unit Commitment,

   Yan, Bing ; Luh, Peter B. ; Litvinov, Eugene ; Zheng, Tongxin ; Schiro, Dane ; Bragin, Mikhail A. ; Zhao, Feng ; Zhao, Jinye ; Lelic, Izudin ( August 2019 , IEEE Power & Energy Society General Meeting)

   Unit Commitment is an important problem faced by independent system operators. It is usually formulated as a Mixed Binary Linear Programming (MBLP) problem, and is believed to be NP hard. To solve UC problems efficiently, an idea is through formulation tightening. If constraints can be transformed to directly delineate an MBLP problem’s convex hull during data preprocessing, then the problem can be solved by using linear programming methods. The resulting formulation can be reused for other data sets, tremendously reducing computational requirements. To achieve the above goal, both unit- and system-level constraints are tightened with synergistic combination in this paper. Unit-level constraints are tightened based on existing cuts and novel “constraint-and-vertex conversion” and vertex projection processes. To tighten system-level constraints, selected cuts are applied and some potentially powerful cuts are identified. Numerical results demonstrate the effectiveness of tightening unit- and system-level constraints. 

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5. Ordered Tree Decomposition for HRG Rule Extraction

   https://doi.org/10.1162/coli_a_00350  

   Gildea, Daniel ; Satta, Giorgio ; Peng, Xiaochang ( June 2019 , Computational Linguistics)

   null (Ed.)

   We present algorithms for extracting Hyperedge Replacement Grammar (HRG) rules from a graph along with a vertex order. Our algorithms are based on finding a tree decomposition of smallest width, relative to the vertex order, and then extracting one rule for each node in this structure. The assumption of a fixed order for the vertices of the input graph makes it possible to solve the problem in polynomial time, in contrast to the fact that the problem of finding optimal tree decompositions for a graph is NP-hard. We also present polynomial-time algorithms for parsing based on our HRGs, where the input is a vertex sequence and the output is a graph structure. The intended application of our algorithms is grammar extraction and parsing for semantic representation of natural language. We apply our algorithms to data annotated with Abstract Meaning Representations and report on the characteristics of the resulting grammars. 

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