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1. Induced subgraphs and tree decompositions IX. Grid theorem for perforated graphs

   https://doi.org/10.19086/aic.2025.3  

   Alecu, Bogdan; Chudnovsky, Maria; Spirkl, Sophie (March 2025, Advances in Combinatorics)

   The celebrated Erdős-Pósa Theorem, in one formulation, asserts that for every c ∈ N, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of c cycles have bounded treewidth. What can we say about the treewidth of graphs containing no induced subgraph isomorphic to the disjoint union of c cycles? Let us call these graphs c-perforated. While 1-perforated graphs have treewidth one, complete graphs and complete bipartite graphs are examples of 2-perforated graphs with arbitrarily large treewidth. But there are sparse examples, too: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek constructed 2-perforated graphs with arbitrarily large treewidth and no induced subgraph isomorphic to K3 or K3,3; we call these graphs occultations. Indeed, it turns out that a mild (and inevitable) adjustment of occultations provides examples of 2-perforated graphs with arbitrarily large treewidth and arbitrarily large girth, which we refer to as full occultations. Our main result shows that the converse also holds: for every c ∈ N, a c-perforated graph has large treewidth if and only if it contains, as an induced subgraph, either a large complete graph, or a large complete bipartite graph, or a large full occultation. This distinguishes c-perforated graphs, among graph classes purely defined by forbidden induced subgraphs, as the first to admit a grid-type theorem incorporating obstructions other than subdivided walls and their line graphs. More generally, for all c, o ∈ N, we establish a full characterization of induced subgraph obstructions to bounded treewidth in graphs containing no induced subgraph isomorphic to the disjoint union of c cycles, each of length at least o + 2. 

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2. Induced subgraphs and tree decompositions XII. Grid theorem for pinched graphs

   https://doi.org/10.5802/igt.6  

   Alecu, Bogdan; Chudnovsky, Maria; Hajebi, Sepehr; Spirkl, Sophie (January 2025, Innovations in Graph Theory)

   Given c ∈ N, we say a graph G is c-pinched if G does not contain an induced subgraph consisting of c cycles, all going through a single common vertex and otherwise pairwise disjoint and with no edges between them. What can be said about the structure of c-pinched graphs? For instance, 1-pinched graphs are exactly graphs of treewidth 1. However, bounded treewidth for c > 1 is immediately seen to be a false hope because complete graphs, complete bipartite graphs, subdivided walls and line graphs of subdivided walls are all examples of 2-pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of c, discovered by Pohoata and later independently by Davies, consisting of 3-pinched graphs with unbounded treewidth and no large induced subgraph isomorphic to any of the first four obstructions. We fuse the above five examples into a grid-type theorem fully describing the unavoidable induced subgraphs of pinched graphs with large treewidth. More precisely, we prove that for every c ∈ N, a c-pinched graph G has large treewidth if and only if G contains one of the following as an induced subgraph: a large complete graph, a large complete bipartite graph, a subdivision of a large wall, the line graph of a subdivision of a large wall, or a large graph from the Pohoata-Davies construction. Our main result also generalizes to an extension of pinched graphs where the lengths of excluded cycles are lower-bounded. 

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3. Fast Counting and Utilizing Induced 6-Cycles in Bipartite Networks

   https://doi.org/10.1109/TKDE.2025.3546516  

   Niu, Jason; Zola, Jaroslaw; Sarıyüce, Ahmet Erdem (June 2025, IEEE Transactions on Knowledge and Data Engineering)

   Bipartite graphs are a powerful tool for modeling the interactions between two distinct groups. These bipartite relationships often feature small, recurring structural patterns called motifs which are building blocks for community structure. One promising structure is the induced 6-cycle which consists of three nodes on each node set forming a cycle where each node has exactly two edges. In this paper, we study the problem of counting and utilizing induced 6-cycles in large bipartite networks. We first consider two adaptations inspired by previous works for cycle counting in bipartite networks. Then, we introduce a new approach for node triplets which offer a systematic way to count the induced 6-cycles, used in BATCHTRIPLETJOIN. Our experimental evaluation shows that BATCHTRIPLETJOIN is significantly faster than the other algorithms while being scalable to large graph sizes and number of cores. On a network with 112M edges, BATCHTRIPLETJOIN is able to finish the computation in 78 mins by using 52 threads. In addition, we provide a new way to identify anomalous node triplets by comparing and contrasting the butterfly and induced 6-cycle counts of the nodes. We showcase several case studies on real-world networks from Amazon Kindle ratings, Steam game reviews, and Yelp ratings. 

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4. Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

   https://doi.org/10.19086/aic.2022.6  

   Chudnovsky, Maria; Abrishami, Tara; Hajebi, Sepehr; Spirkl, Sophie (September 2022, Advances in Combinatorics)

   A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-_free_ if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $$c$$ for which every (theta, triangle)-free graph $$G$$ has treewidth at most $$c\log (|V(G)|)$$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $$G$$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $$k$$-Coloring (for fixed $$k$$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph. 

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5. Point-hyperplane Incidence Geometry and the Log-rank Conjecture

   https://doi.org/10.1145/3543684  

   Singer, Noah; Sudan, Madhu (June 2022, ACM Transactions on Computation Theory)

   We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in ℝ d that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2 –polylog( d ) ) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words, the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank- d matrix containing at most O (1) distinct entries in each column contains a submatrix of fractional size 2 –polylog( d ) , in which each column is constant. We prove that our conjecture is equivalent to the log-rank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k -partitions to bounds for parallel ( k -1)-partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density Ω (ε 2 d / d ) in any d -dimensional configuration with incidence density ε, qualitatively matching previous results proved using sophisticated geometric techniques. We also improve an upper-bound construction of Apfelbaum and Sharir [ 2 ], yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is Ω (1/√ d ). Finally, we discuss various constructions (due to others) of products of Boolean matrices which yield configurations with incidence density Ω (1) and complete bipartite subgraph density 2 -Ω (√ d ) , and pose several questions for this special case in the alternative language of extremal set combinatorics. Our framework and results may help shed light on the difficulty of improving Lovett’s Õ(√ rank( f )) bound [ 20 ] for the log-rank conjecture. In particular, any improvement on this bound would imply the first complete bipartite subgraph size bounds for parallel 3-partitioned configurations which beat our generic bounds for unstructured configurations. 

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