Beyond Helly graphs: the diameter problem on absolute retracts

Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some graph class ${\cal C}$ if it is a retract of any $G \in {\cal C}$ of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized $\tilde{\cal O}(m\sqrt{n})$ time. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of $k$-chromatic graphs, for every fixed $k \geq 3$. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively

Topological Methods for Social and Behavioral Systems

Methods based in algebra and geometry are introduced for the mathematical formulation of problems in the social and behavioral sciences. Specifically, the paper introduces the main concepts of singularity theory, catastrophe theory and q-analysis for the characterization of the global structure of social systems. Applications in urban land development, electric power generation and international conflict are given to illustrate the methodology. The paper concludes with an outline for a general mathematical theory of surprises, together with a program for investigating the systemic property of resilience

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Role of Shape in the Self-Assembly of Anisotropic Colloids.

Self-assembly is the process of spontaneous organization of a set of interacting components. We examine how particle shape drives the self-assembly of colloids in three different systems. When particles interact only via their shape, entropic crystallization can occur; we discuss a design strategy using the Voronoi tesslelation to create “Voronoi particles,” (VP) which are hard particles in the shape of Voronoi cells of their target structure. Although VP stabilize their target structure in the limit of infinite pressure, the self-assembly of the same structure at moderate pressure is not guaranteed. We find that more symmetric crystals are often preferred due to entropic contributions of several kBT from configurational degeneracies. We characterize the assembly of VP in terms of their symmetries and the complexities of the target structure and demonstrate how controlling the degeneracies through modifying shape and field-directed assembly can improve the assembly propensity. With the addition of non-adsorbing, polymers, hard colloids experience an attraction dependent on polymer concentration, the form of which is dictated by the colloid shape; we study a system of oblate, spheroidal colloids that self-assemble thread-like clusters. In both simulation and experiment the colloids condense into disordered droplets at low polymer concentrations; at higher concentrations we observe kinetic arrest into primarily linear clusters of aligned colloids. We show that the mechanical stabilty of these low-valence structures results from the anisotropic particle shape. Particle surfaces can be patterned with metal coatings, introducing enthalpic attraction between particles; we study a system of prolate spheroidal colloids, half-coated in gold. We show with experiments and computer simulations that Janus ellipsoids can self-assemble into self-limiting one-dimensional fibers with shape-memory properties, and that the fibrillar assemblies can be actuated on application of an external alternating-current electric field. Actuation of the fibers occurs through a sliding mechanism (allowed by the curved ellipsoidal surface) that permits the reversible elongation of the Janus-ellipsoid chains by ~36%. In each case, we find shape plays a critical role. By understanding and isolating its impact, we enhance shape's utility as a parameter for the design of self-assembling colloids.PhDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111630/1/baschult_1.pd