Geometric Graphs: Turán Problems

On grids in topological graphs

The max-bisection and min-bisection problems are to find a partition of the vertices of a graph into two equal size subsets that, respectively, maximizes or minimizes the number of edges with endpoints in both subsets. We design the first polynomial time approximation scheme for the max-bisection problem on arbitrary planar graphs solving a long-standing open problem. The method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs, in particular max- and min-bisection, which could be of independent interest. Using a similar method we design also the first polynomial time approximation scheme for max-bisection on unit disk graphs (which could also be easily extended to other geometrically defined graphs).

We propose a radically new family of geometric graphs, i.e., Hypocomb, Reduced Hypocomb and Local Hypocomb. The first two are extracted from a complete graph; the last is extracted from a Unit Disk Graph (UDG). We analytically study their properties including connectivity, planarity and degree bound. All these graphs are connected (provided the original graph is connected) planar. Hypocomb has unbounded degree while Reduced Hypocomb and Local Hypocomb have maximum degree 6 and 8, respectively. To our knowledge, Local Hypocomb is the first strictly-localized, degree-bounded planar graph computed using merely 1-hop neighbor position information. We present a construction algorithm for these graphs and analyze its time complexity. Hypocomb family graphs are promising for wireless ad hoc networking. We report our numerical results on their average degree and their impact on FACE routing. We discuss their potential applications and some open problems.

On asymptotic packing of geometric graphs

Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

Max point-tolerance graphs

Localization game on geometric and planar graphs

Drug-drug interaction (DDI) can trigger many adverse effects in patients and has emerged as a threat to medicine and public health. Therefore, it is important to predict potential drug interactions since it can provide combination strategies of drugs for systematic and effective treatment. Existing deep learning-based methods often rely on DDI functional networks, or use them as an important part of the model information source. However, it is difficult to discover the interactions of a new drug. To address the above limitations, we propose a geometric molecular graph representation learning model (Mol-DDI) for DDI prediction based on the basic assumption that structure determines function. Mol-DDI only considers the covalent and non-covalent bond information of molecules, then it uses the pre-training idea of large-scale models to learn drug molecular representations and predict drug interactions during the fine-tuning process. Experimental results show that the Mol-DDI model outperforms others on the three datasets and performs better in predicting new drug interaction experiments.

Predicting protein-ligand binding affinities is a critical problem in drug discovery and design. A majority of existing methods fail to accurately characterize and exploit the geometrically invariant structures of protein-ligand complexes for predicting binding affinities. In this study, we propose Geo-protein-ligand binding affinity (PLA), a geometric equivariant graph representation learning framework with local-global structure awareness, to predict binding affinity by capturing the geometric information of protein-ligand complexes. Specifically, the local structural information of 3-D protein-ligand complexes is extracted by using an equivariant graph neural network (EGNN), which iteratively updates node representations while preserving the equivariance of coordinate transformations. Meanwhile, a graph transformer is utilized to capture long-range interactions among atoms, offering a global view that adaptively focuses on complex regions with a significant impact on binding affinities. Furthermore, the multiscale information from the two channels is integrated to enhance the predictive capability of the model. Extensive experimental studies on two benchmark datasets confirm the superior performance of Geo-PLA. Moreover, the visual interpretation of the learned protein-ligand complexes further indicates that our model offers valuable biological insights for virtual screening and drug repositioning.

The complete twisted graph of order n, denoted by $$T_n$$ , is a complete simple topological graph with vertices $$u_1, u_2, \ldots , u_n$$ such that two edges $$u_iu_j$$ and $$u_{i'}u_{j'}$$ cross if and only if $$ i<i'<j'<j $$ or $$ i'<i<j<j' $$ . The convex geometric complete graph of order n, denoted by $$G_n$$ , is a convex geometric graph with vertices $$v_1, v_2, \ldots , v_n$$ placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set $$\{v_1,v_2,\ldots , v_n \}$$ having the property of being planar when embedded in both $$T_n$$ and $$G_n$$ . Given a connected graph G the (combinatorial) tree graph $$\mathcal {T}(G)$$ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in $$\mathcal {T}(G)$$ if there are edges $$e \in P$$ and $$f\in Q$$ such that $$Q=P - e + f$$ . For all positive integers n, we denote by $$\mathcal {T}(n)$$ the graph $$\mathcal {T}(K_n)$$ . The biplanar tree graph, $$\mathcal {B}(n)$$ , is the subgraph of $$\mathcal {T}(n)$$ induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.

NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

Let G be a geometric graph whose vertex set S is a set of n points in ℝ d . The stretch factor of two distinct points p and q in S is the ratio of their shortest-path distance in G and their Euclidean distance. We consider the problem of approximating the average of the n choose 2 stretch factors determined by all pairs of points in S . We show that for paths, cycles, and trees, this average can be approximated, within a factor of 1+e, in O ( n polylog( n )) time. For plane graphs in ℝ 2 , we present a (2+e)-approximation algorithm with running time O ( n 5/3 polylog( n )), and a (4+e)-approximation algorithm with running time O ( n 3/2 polylog( n )). Finally, we show that, for any tree in ℝ 2 , the exact average of the squares of the n choose 2 stretch factors can be computed in O ( n 11/6 ) time.

We consider the following problem which is motivated by two different contexts independently, namely graph theory and combinatorial optimization.Given a 3-connected planar graph G with n vertices, is there a spanning closed walk W with at most 4n/3 edges?In graph theory, the above question is motivated by the famous hamiltonian result by Tutte in 1956 which says that every 4-connected planar graph is hamiltonian (a simpler proof is given by Thomassen in 1983). What happens if we relax the 4-connectivity? There is a 3-connected planar graph that is not hamiltonian, but how about a spanning close walk (which is exactly a traveling salesman tour. Sometimes such a walk is called hamiltonian walk)? How many edges are necessary to cover all the vertices of a 3-connected planar graph by a closed walk? This is exactly the above question.In combinatorial optimization, the famous traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems. In spite of a vast amount of research several important questions remain open. In particular, the best known upper bound is not believed to be best possible. A promising direction to improve this approximation guarantee, has long been to understand the power of a linear program known as the Held-Karp relaxation [11]. On the one hand, the best lower bound on its integrality gap (for the symmetric case) is 4/3 and indeed the famous (so called 4/3-)conjecture said that this lower bound would be tight [10]. Goemans pointed out that there is a planar graph that achieves this bound. So he brought attention to the above question, i.e, the famous 4/3-conjecture is always true for 3-connected planar graphs.We prove the above problem in the following strong form;Given a circuit graph (which is obtained from a 3-connected planar graph by deleting one vertex) with n vertices, there is a spanning closed walk with at most 4(n − 1)/3 edges such that each edge is used at most twice.Moreover, our proof is constructive (and purely combinatorial) in a sense that there is an O(n2) algorithm to construct, given a 3-connected planar graph, such a walk.We shall construct an example that shows that the bound 4(n − 1)/3 is essentially tight. We also point out that 2-connected planar graphs may not have such a walk, as K2,n−2 shows.

Transformer models have great potential in Graph Representation Learning (GRL) for efficiently scaling the learning process on large datasets and solving many challenges presented in Graph Neural Networks, e.g., oversmoothing and suspended animation. To represent each node of a graph, Transformer models as input usually take a node together with the node context, i.e., a set of other nodes that serve as learning context for the target node. However, current GRL Transformer models mainly consider the graph topology when selecting the node context for each target node. In this work, we demonstrate the important role of node features in selecting the node context. Specifically, we propose a hybrid approach for selecting node context that considers both the graph topology and the semantic similarities between node features. Through the empirical evaluations, we show the advantages of our hybrid node context selection method for a downstream classification task on various datasets compared to selection methods that only consider graph topology or semantic similarities. The best classification accuracy improvements of our proposed hybrid methods over the baseline methods on each dataset range from 0.77% to 6.05%.

The M‐degree of an edge xy in a graph is the maximum of the degrees of x and y. The M‐degree of a graph G is the minimum over M‐degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4‐cycles has M‐degree at most 8 and gave an example of such a graph with M‐degree 3. This yields upper bounds on the game chromatic number of C4‐free planar graphs. We determine the maximum possible M‐degrees for planar, projective‐planar and toroidal graphs without leaves and 4‐cycles. In particular, for planar and projective‐planar graphs this maximum is 7. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 80–85, 2009