Unit Grid Intersection Graphs: Recognition and Properties

It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs (UGIGs), the recognition remains hard, as well as for all graph classes contained inbetween. The result holds even when considering only graphs with arbitrarily large girth. Furthermore, we ask the question of representing UGIGs on grids of minimal size. We show that the UGIGs that can be represented in a square of side length 1+epsilon, for a positive epsilon no greater than 1, are exactly the orthogonal ray graphs, and that there exist families of trees that need an arbitrarily large grid

Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination

Space-Time Circuit-to-Hamiltonian Construction and Its Applications

The circuit-to-Hamiltonian construction translates dynamics (a quantum circuit and its output) into statics (the groundstate of a circuit Hamiltonian) by explicitly defining a quantum register for a clock. The standard Feynman-Kitaev construction uses one global clock for all qubits while we consider a different construction in which a clock is assigned to each interacting qubit. This makes it possible to capture the spatio-temporal structure of the original quantum circuit into features of the circuit Hamiltonian. The construction is inspired by the original two-dimensional interacting fermionic model (see http://link.aps.org/doi/10.1103/PhysRevA.63.040302) We prove that for one-dimensional quantum circuits the gap of the circuit Hamiltonian is appropriately lower-bounded, partially using results on mixing times of Markov chains, so that the applications of this construction for QMA (and partially for quantum adiabatic computation) go through. For one-dimensional quantum circuits, the dynamics generated by the circuit Hamiltonian corresponds to diffusion of a string around the torus.Comment: 27 pages, 5 figure