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A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of…
This paper studies real-world road networks from an algorithmic perspective, focusing on empirical studies that yield useful properties of road networks that can be exploited in the design of fast algorithms that deal with geographic data.…
Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes…
A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
We propose a statistical model for graphs with a core-periphery structure. To do this we define a precise notion of what it means for a graph to have this structure, based on the sparsity properties of the subgraphs of core and periphery…
Optimal paths connecting randomly selected network nodes and fixed routers are studied analytically in the presence of non-linear overlap cost that penalizes congestion. Routing becomes increasingly more difficult as the number of selected…
A hypergraph is a generalization of a graph, in which a hyperedge can connect multiple vertices, modeling complex relationships involving multiple vertices simultaneously. Hypergraph pattern matching, which is to find all isomorphic…
We consider the number of crossings in a graph which is embedded randomly on a convex set of points. We give an estimate to the normal distribution in Kolmogorov distance which implies a convergence rate of order $n^{-1/2}$ for various…
Graph spanners are sparse subgraphs which approximately preserve all pairwise shortest-path distances in an input graph. The notion of approximation can be additive, multiplicative, or both, and many variants of this problem have been…
Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings,…
Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ.…
Complex systems of interacting components often can be modeled by a simple graph $\mathcal{G}$ that consists of a set of $n$ nodes and a set of $m$ edges. Such a graph can be represented by an adjacency matrix $A\in\R^{n\times n}$, whose…
We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…
Compound graphs are networks in which vertices can be grouped into larger subsets, with these subsets capable of further grouping, resulting in a nesting that can be many levels deep. In several applications, including biological workflows,…
Graph embedding is a transformation of nodes of a network into a set of vectors. A good embedding should capture the underlying graph topology and structure, node-to-node relationship, and other relevant information about the graph, its…
Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181]…
We investigate edge-intersection graphs of paths in the plane grid, regarding a parameter called the bend-number. I.e., every vertex is represented by a grid path and two vertices are adjacent if and only if the two grid paths share at…
A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…
We prove that all $1$-vertex spatial graphs with adequate diagrams have minimal crossing number, and that spatial graph diagrams obtained by replacing vertices and edges of a planar embedded graph by minimal crossing link or spatial graph…