arXiv research

Characterizes minor-minimal separating projective planar graphs and their generalizations.

New IPL graphs identified and conditions for their projective embeddings established.

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented s…

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…

In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

Study examines how changing regions affects planar graphs.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

Study on planar graph braid groups' second homology.

New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.

The complement of a non-separating planar graph contains a K_n minor.

Study on planar graphs in Poincare model of hyperbolic geometry.

String graphs are closely related to planar graphs in terms of distances.

The study extends Tutte's conflict graph concept to nonplanar graphs.

Spatial graphs are decomposed into planar forests and braids.

In his 1930 paper, Kuratowksi categorized planar graphs, proving that a finite graph $Γ$ is planar if and only if it does not contain a subgraph that is homeomorphic to $K_5$, the complete graph on 5 vertices, or $K_{3,3}$, the complete bipartite graph on six vertices. In their 2001 paper, Davis and Okun point out that…

Proves planar graphs' configuration spaces have highest topological complexity.

A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\forall v$…

Researchers compute c-projective symmetry algebras for Kähler surfaces.

Sharp bounds for spanning tree entropy in planar lattices.

Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.

Study classifies Halin graphs with positive curvature.

The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful p…

Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.

We describe a new technique for computing lower-bounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are represented in terms of …

Asymptotic dimension of planes and graphs is at most three.

Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.

Study of higher-dimensional contact manifolds and their properties.

We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $λ$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize …

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three variables, $A$, $B$ and $a$, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=…

Approximates cycles in planar and bounded-genus graphs.

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web …

We prove that the spectral gap of a finite planar graph $X$ is bounded by $λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Cur…

New method realizes planar graphs as Reeb graphs of algebraic functions.

We present old and recent results on rank problems and linearizability of geodesic planar webs.

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is …

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tre…

Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a m…

We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995…

Origamis' orbits are non-planar except for a few specific cases.

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…

Moduli space linked to Tait colorings of planar graphs.

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose ele…

Study on linking numbers in random book embeddings of complete graphs.