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Papers by Jo Ellis-Monaghan

Symmetry

Motivated by the problem arising out of DNA origami, we give a general counting framework and enu... more Motivated by the problem arising out of DNA origami, we give a general counting framework and enumeration formulas for various cellular embeddings of bouquets and dipoles under different kinds of symmetries. Our algebraic framework can be used constructively to generate desired symmetry classes, and we use Burnside’s lemma with various symmetry groups to derive the enumeration formulas. Our results assimilate several existing formulas into this unified framework. Furthermore, we provide new formulas for bouquets with colored edges (and thus for bouquets in nonorientable surfaces) as well as for directed embeddings of directed bouquets. We also enumerate vertex-labeled dipole embeddings. Since dipole embeddings may be represented by permutations, the formulas also apply to certain equivalence classes of permutations and permutation matrices. The resulting bouquet and dipole symmetry formulas enumerate structures relevant to a wide variety of areas in addition to DNA origami, includin...

We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) t... more We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the canonical identification of a graph's edges with those of its dual, petrial, or trial, whereas the more natural definition allows any isomorphism. Here we define a new ribbon group action on ribbon graphs, using a semidirect product of the original ribbon group with a permutation group, to take (partial) twists and duals of ribbon graphs while also encoding graph isomorphisms. This brings new algebraic tools to bear on the natural definitions of self-duality etc., as a ribbon graph is a fixed point of this new ribbon group action exactly when it is isomorphic to one of its (partial) twisted duals. With these tools, we prove that every ribbon graph has in its orbit an orientable embedded bouquet, whose (partial) twisted duality properties propagate th...

Dagstuhl Reports, 2019

This report documents the programme and outcomes of Dagstuhl Seminar 19401 "Comparative Theory fo... more This report documents the programme and outcomes of Dagstuhl Seminar 19401 "Comparative Theory for Graph Polynomials". The study of graph polynomials has become increasingly active, with new applications and new graph polynomials being discovered each year. The genera of graph polynomials are diverse, and their interconnections are rich. Experts in the field are finding that proof techniques and results established in one area can be successfully extended to others. From this a general theory is emerging that encapsulates the deeper interconnections between families of graph polynomials and the various techniques, computational approaches, and methodologies applied to them. The overarching aim of this Seminar was to exploit commonalities among polynomial invariants of graphs, matroids, and related combinatorial structures. Model-theoretic, computational and other methods were used in order to initiate a comparative theory that collects the current state of knowledge into a more cohesive and powerful framework.

Branched junction molecule assembly of DNA nanostructures, pioneered by Seeman’s laboratory in th... more Branched junction molecule assembly of DNA nanostructures, pioneered by Seeman’s laboratory in the 1980s, has become increasingly sophisticated, as have the assembly targets. A critical design step is finding minimal sets of branched junction molecules that will self-assemble into target structures without unwanted substructures forming. We use graph theory, which is a natural design tool for self-assembling DNA complexes, to address this problem. After determining that finding optimal design strategies for this method is generally NP-complete, we provide pragmatic solutions in the form of programs for special settings and provably optimal solutions for natural assembly targets such as platonic solids, regular lattices, and nanotubes. These examples also illustrate the range of design challenges.

Dagstuhl Reports, 2016

This report documents the program and the outcomes of Dagstuhl Seminar 16241 "Graph Polynomi... more This report documents the program and the outcomes of Dagstuhl Seminar 16241 "Graph Polynomials: Towards a Comparative Theory". The area of graph polynomials has become in recent years incredibly active, with new applications and new graph polynomials being discovered each year. However, the resulting plethora of techniques and results now urgently requires synthesis. Beyond catalogues and classifications we need a comparative theory. The intent of this 5-day Seminar was to further a general theory of graph polynomials.

Combinatorics, Probability and Computing, 2021

We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutat... more We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the...

Journal of Knot Theory and Its Ramifications, 2020

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origam... more Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular...

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials,... more A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and 1