Cyclically Orientable Graphs

Barot, Geiss and Zelevinsky define a notion of a ``cyclically orientable graph'' and use it to devise a test for whether a cluster algebra is of finite type. Barot, Geiss and Zelivinsky's work leaves open the question of giving an efficient characterization of cyclically orientable graphs. In this paper, we give a simple recursive description of cyclically orientable graphs, and use this to give an O(n) algorithm to test whether a graph on $n$ vertices is cyclically orientable. Shortly after writing this paper, I learned that most of its results had been obtained independently by Gurvich; I am placing this paper on the arXiv to spread knowledge of these results

11-perfectly orientable K4K_4-minor-free and outerplanar graphs

A graph $G$ is said to be $1$-perfectly orientable if it has an orientation such that for every vertex $v\in V(G)$, the out-neighborhood of $v$ in $D$ is a clique in $G$. In $1982$, Skrien posed the problem of characterizing the class of $1$-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to $2$-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of $1$-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of $1$-perfectly orientable $K_4$-minor-free graphs and of $1$-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of $2$-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most~$2$

Polynomial recognition of cluster algebras of finite type

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type. Using the standard definition, the problem is infeasible since it uses mutations that can lead to an infinite process. Barot, Geiss and Zelevinsky (2006) presented an easier way to verify if a given algebra is of finite type, by testing that all chordless cycles of the graph related to the algebra are cyclically oriented and that there exists a positive quasi-Cartan companion of the skew-symmetrizable matrix related to the algebra. We develop an algorithm that verifies these conditions and decides whether or not a cluster algebra is of finite type in polynomial time. The second part of the algorithm is used to prove that the more general problem to decide if a matrix has a positive quasi-Cartan companion is in NP.Comment: 14 page

Orientable embeddings and orientable cycle double covers of projective-planar graphs

In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2-edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.Comment: 16 pages, 3 figure

Cluster algebras of finite type and positive symmetrizable matrices

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.Comment: 20 pages. In version 3, some new material is added in the end of section 2, discussing the classification and characterizations of positive quasi-Cartan matrices. In final version 4, Proposition 2.9 is corrected and its proof expanded. To appear in J. London Math. Soc. In version 5 only typos in the arXiv data fixe

Embeddings of 3-connected 3-regular planar graphs on surfaces of non-negative Euler characteristic

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular planar graph $G$ embedded on the projective-plane, the torus and the Klein bottle, and give a one-to-one correspondence between inequivalent embeddings of $G$ on each surface and some subgraphs of the dual of $G$ embedded on the sphere. These results enable us to give explicit bounds for the number of inequivalent embeddings of $G$ on each surface, and propose effective algorithms for enumerating and counting these embeddings.Comment: 19 pages, 12 figure

Translation equivalence in free groups

Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements $g,h$ in a free group $F$ have the property that for every free isometric action of $F$ on an $\mathbb{R}$-tree $X$ the translation lengths of $g$ and $h$ on $X$ are equal. We give a combinatorial characterization of this phenomenon, called translation equivalence, in terms of Whitehead graphs and exhibit two difference sources of it. The first source of translation equivalence comes from representation theory and $SL_2$ trace identities. The second source comes from geometric properties of groups acting on real trees and a certain power redistribution trick. We also analyze to what extent these are applicable to the tree actions of surface groups that occur in the Thurston compactification of the Teichmuller space.Comment: revised version, to appear in Transact. Amer. Math. Soc.; two .eps figure

How to Uncross Some Modular Metrics

Let $\mu$ be a metric on a set T, and let c be a nonnegative function on the unordered pairs of elements of a superset $V\supseteq T$. We consider the problem of minimizing the inner product $c\cdot m$ over all semimetrics m on V such that m coincides with $\mu$ within T and each element of V is at zero distance from T (a variant of the {\em multifacility location problem}). In particular, this generalizes the well-known multiterminal multiway) cut problem. Two cases of metrics $\mu$ have been known for which the problem can be solved in polynomial time: (a) $\mu$ is a modular metric whose underlying graph $H(\mu)$ is hereditary modular and orientable (in a certain sense); and (b) $\mu$ is a median metric. In the latter case an optimal solution can be found by use of a cut uncrossing method. \Xcomment{We give a common generalization for both cases by proving that the problem is in P for any modular metric $\mu$ whose all orbit graphs are hereditary modular and orientable. To this aim, we show the existence of a retraction of the Cartesian product of the orbit graphs to $H(\mu)$, which enables us to elaborate an analog of the cut uncrossing method for such metrics $\mu$.} In this paper we generalize the idea of cut uncrossing to show the polynomial solvability for a wider class of metrics $\mu$, which includes the median metrics as a special case. The metric uncrossing method that we develop relies on the existence of retractions of certain modular graphs. On the negative side, we prove that for $\mu$ fixed, the problem is NP-hard if $\mu$ is non-modular or $H(\mu)$ is non-orientable.Comment: 25 page

Positive Grassmannian and polyhedral subdivisions

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian.Comment: 25 page

Virtual Geometricity is Rare

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.Comment: 8 pages, 2 figures v2 adds a link to the computer scripts used in the paper; v3 13pages, to appear in LMS Journal of Computation and Mathematic