Metric graph theory and geometry: a survey

Abstract

The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l 1-graphs. Several kinds of l 1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆-)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or tree-like graphs such as distance-hereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the graph classes reported in this survey.

Origin of Hyperbolicity in Brain-to-Brain Coordination Networks

“…In the clique complex method [47,48] that we use here, the elementary geometrical descriptors of the network structure are identified as cliques of different orders q = 0, 1, 2 · · · q max , i.e., nodes, links, triangles, tetrahedra and higher-order cliques up to the largest size q max + 1 that occurs in the network. Moreover, the method identifies the nodes that make a particular clique, which allows determining the ways that different cliques interconnect with each other via shared faces-cliques of the lower order, to form a simplicial complex.…”

Topological Characterization of Complex Systems: Using Persistent Entropy

“…We use the Bron-Kerbosch algorithm [20] for listing all of the maximal cliques of a graph. As shown in Figure 3, a simplicial clique complex of G is the simplicial complex whose simplices are all of the maximal cliques of G. Since a k-clique is equivalent to a (k − 1)-simplicial complex, then a 3-clique is equivalent to a 2-simplicial complex, and so on [21]. The computational algorithms for the characterization of simplicial complexes require as input a filtered simplicial complex, i.e., a simplicial complex equipped with a filter value.…”

Recognition of $C_4$-Free and 1/2-Hyperbolic Graphs

“…There is a linear-time reduction from the C 4 -free graph recognition problem to the problem of deciding whether a graph is 1 2 -hyperbolic. 2 The notationÕ(f (n)) is for a complexity f (n) · log O(1) n. Proof. Let G = (V, E) be an instance of the C 4 -free graph recognition problem.…”

“…An illustration of the construction of G [2] is presented in Figure 4. It might help to observe that for every edge of G 2 i.e., for every two distinct vertices u, v such that d G (u, v) ≤ 2, there are exactly two corresponding edges in G [2] , denoted by {(u, 0), (v, 1)} and {(u, 1), (v, 0)}, that connect the sets V × {0} and V × {1}.…”