Lower Bounds for Electrical Reduction on Surfaces

We strengthen the connections between electrical transformations and homotopy from the planar setting - observed and studied since Steinitz - to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the number of electrical transformations required to reduce an n-vertex graph on surface in the worst case [SOCG 2016] in two different directions. Our previous Omega(n^{3/2}) lower bound applies only to facial electrical transformations on plane graphs with no terminals. First we provide a stronger Omega(n^2) lower bound when the planar graph has two or more terminals, which follows from a quadratic lower bound on the number of homotopy moves in the annulus. Our second result extends our earlier Omega(n^{3/2}) lower bound to the wider class of planar electrical transformations, which preserve the planarity of the graph but may delete cycles that are not faces of the given embedding. This new lower bound follow from the observation that the defect of the medial graph of a planar graph is the same for all its planar embeddings

Transformaciones ∆ − Y en redes

La Combinatoria se compone de varias ramas que involucran el estudio de procesos finitos y estructuras discretas. Las gráficas son estructuras que constituyen el concepto central del estudio de una rama muy robusta de la Combinatoria. Así que resulta de gran interés investigar sus propiedades, invariantes y clasificaciones en familias que comparten propiedades. Un conjunto de resultados en teoría de graficas muestran que una clase de graficas pueden ser reducidas a una forma canónica mediante la aplicación de ciertas operaciones. Se puede demostrar, que la aplicación inversa de esas mismas operaciones puede generar todas las gráficas en una clase. Uno de los conjuntos más importantes de reducciones usadas en la teoría de graficas son las reducciones serie-paralelo, al aplicar una reducción de este tipo a una gráfica, se disminuye el número de sus aristas. Las operaciones que no alteran el número de aristas de la gráfica se llaman transformaciones, en particular y como centro de análisis en este trabajo, se estudian las transformaciones ∆ − Y

Tightening curves and graphs on surfaces

Any continuous deformation of closed curves on a surface can be decomposed into a finite sequence of local changes on the structure of the curves; we refer to such local operations as homotopy moves. Tightening is the process of deforming given curves into their minimum position; that is, those with minimum number of self-intersections. While such operations and the tightening process has been studied extensively, surprisingly little is known about the quantitative bounds on the number of homotopy moves required to tighten an arbitrary curve. An unexpected connection exists between homotopy moves and a set of local operations on graphs called electrical transformations. Electrical transformations have been used to simplify electrical networks since the 19th century; later they have been used for solving various combinatorial problems on graphs, as well as applications in statistical mechanics, robotics, and quantum mechanics. Steinitz, in his study of 3-dimensional polytopes, looked at the electrical transformations through the lens of medial construction, and implicitly established the connection to homotopy moves; later the same observation has been discovered independently in the context of knots. In this thesis, we study the process of tightening curves on surfaces using homotopy moves and their consequences on electrical transformations from a quantitative perspective. To derive upper and lower bounds we utilize tools like curve invariants, surface theory, combinatorial topology, and hyperbolic geometry. We develop several new tools to construct efficient algorithms on tightening curves and graphs, as well as to present examples where no efficient algorithm exists. We then argue that in order to study electrical transformations, intuitively it is most beneficial to work with monotonic homotopy moves instead, where no new crossings are created throughout the process; ideas and proof techniques that work for monotonic homotopy moves should transfer to those for electrical transformations. We present conjectures and partial evidence supporting the argument