Symmetric tangled Platonic polyhedra

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols Ih, Oh, and Td). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θz” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

Method for Orthogonal Edge Routing of Directed Layered Graphs with Edge Crossings Reduction

This paper presents a method for automated orthogonal edge routing of directed layered graphs using the described edge crossings reduction heuristic algorithm. The method assumes the nodes are pre-arranged on a rectangular grid composed of layers across the flow direction and lanes along the flow direction. Both layers and lanes are separated by rectangular areas defined as pipes. Each pipe has associated segment tracks. The edges are represented as orthogonal polylines consisting of line segments and routed along the shortest paths. Each segment is assigned to a pipe and to a segment track in it. The edge crossings reduction uses an iterative algorithm to resolve crossings between segments. Conflicting segments are reassigned to adjacent segment tracks, either by swapping with adjacent segments, or by inserting new tracks and calculating the shortest paths of edges. The algorithm proved to be efficient and was implemented in an interactive graph design tool.

Folds induced by multiple parallel or antiparallel double-helices: (pseudo)knotting of single-stranded RNA

ABSTRACTWe develop tools to explore and catalogue the topologies of knotted or pseudoknotted circular folds due to secondary and tertiary interactions within a closed loop of RNA which generate multiple double-helices due (for example) to strand complementarity. The fold topology is captured by a ‘contracted fold’ which merges helices separated by bulges and removes hairpin loops. Contracted folds are either trivial or pseudoknotted. Strand folding is characterised by a rigid-vertex ‘polarised strand graph’, whose vertices correspond to double-helices and edges correspond to strands joining those helices. Each vertex has a plumbline whose polarisation direction defines the helical axis. That polarised graph has a corresponding circular ribbon diagram and canonical alphanumeric fold label. Key features of the ‘fully-flagged’ fold are the arrangement of complementary domains along the strand, described by a numerical bare fold label, and a pair of binary ‘flags’: a parity flag that specifies the twist in each helix (even or odd half-twists), and an orientation flag that characterises each double-helix as parallel or antiparallel. A simple algorithm is presented to translate an arbitrary fold label into a polarised strand graph. Any embedding of the graph in 3-space is an admissible fold geometry; the simplest embeddings minimise the number of edge-crossings in a planar graph drawing. If that number is zero, the fold lies in one of two classes: (a)-type ‘relaxed’ folds, which contain conventional junctions and (b)-type folds whose junctions are described as meso-junctions in H. Wang and N.C. Seeman, Biochem, vol. 34, pp920-929. (c)-type folds induce polarised strand graphs with edge-crossings, regardless of the planar graph drawing. Canonical fold labelling allows us to sort and enumerate all ‘semi-flagged’ folds with up to six contracted double-helices as windings around the edges of a graph-like fold skeleton, whose cyclomatic number - the ‘fold genus’ - ranges from 0 – 3, resulting in a pair of duplexed strands along each skeletal edge. Those semi-flagged folds admit both even and odd double-helical twists. Appending specific parity flags to those semi-flagged folds gives fully-flagged (a)-type folds, which are also enumerated up to genus-3 cases. We focus on all-antiparallel folds, characteristic of conventional ssRNA and enumerate all distinct (a), (b) and (c)-type folds with up to five double-helices. Those circular folds lead to pseudoknotted folds for linear ssRNA strands. We describe all linear folds derived from (a) or (b)-type circular folds with up to four contracted double-helices, whose simplest cases correspond to so-called H, K and L pseudoknotted folds, detected in ssRNA. Fold knotting is explored in detail, via constructions of so-called antifolds and isomorphic folds. We also tabulate fold knottings for (a) and (b)-type folds whose embeddings minimise the number of edge-crossings and outline the procedure for (c)-type folds. The inverse construction - from a specific knot to a suitable nucleotide sequence - results in a hierarchy of knots. A number of specific alternating knots with up to 10 crossings emerge as favoured fold designs for ssRNA, since they are readily constructed as (a)-type all-antiparallel folds.

On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}

The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).

Improvement on the Crossing Number of Crossing-Critical Graphs

The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

Join Products K2,3 + Cn

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

Linear Operators That Preserve Two Genera of a Graph

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.