Constructive Approach to Automorphism Groups of Planar Graphs

2017

In 1975, Babai characterized which abstract groups can be realized as the automorphism groups of planar graphs. In this paper, we give a more detailed and understandable description of these groups. We describe stabilizers of vertices in connected planar graphs as the class of groups closed under the direct product and semidirect products with symmetric, dihedral and cyclic groups. The automorphism group of a connected planar graph is then obtained as a semidirect product of a direct product of these stabilizers with a spherical group. The formulation of the main result is new and original. Moreover, it gives a deeper in the structure of the groups. As a consequence, automorphism groups of several subclasses of planar graphs, including 2-connected planar, outerplanar, and series-parallel graphs, are characterized. Our approach translates into a quadratic-time algorithm for computing the automorphism group of a planar graph which is the first such algorithm described in detail.

Discrete Applied Mathematics, 2007

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an n-geometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2-and 3-geometric automorphism groups of a given graph. We implement the algorithm using Magma [29] and the experimental results show that our approach is very efficient in practice. We also present a drawing algorithm to display 2-and 3-geometric automorphism groups.

This paper introduces the basic definitions and properties of simple graphs which are mainly covered in [1] and [2]. Each definition and property is supported by examples and diagrams. There are also some basic facts used in this paper which have been demonstrated by other researchers such as [3] and [4]. The main concern and the focus in this paper are on the automorphism groups of some graphs. The final part of this work have been on cubic graphs and the Boolian graph B n. To achieve the main points, the group automorphisms have been applied on the automorphisms of some graphs. The permutation groups played the principle role in the case. This was used to study the nature of the graph automorphisms.

Interval graphs are intersection graphs of closed intervals and circle graphs are intersection graphs of chords of a circle. We study automorphism groups of these graphs. We show that interval graphs have the same automorphism groups as trees, and circle graphs have the same automorphism groups as pseudoforests, which are graphs with at most one cycle in every connected component. Our technique determines automorphism groups for classes with a strong structure of all geometric representations, and it can be applied to other similar graph classes. Our results imply polynomial-time algorithms for computing automorphism groups in terms of group products.

Hacettepe Journal of Mathematics and Statistics, 2017

Let G be a finite group and Aut(G) be the group of automorphisms of G. We associate a graph to a group G and fixed automorphism α of G denoted by Γ α G as follows. The vertex set of Γ α G is G \ Z α (G) and two vertices x, g ∈ G \ Z α (G) are adjacent if [g, x]α = 1 or [x, g]α = 1, where [g, x]α = g −1 x −1 gx α and Z α (G) = {x ∈ G | [g, x]α = 1 for all g ∈ G}. In this paper, we state some basic properties of the graph, like connectivity, diameter, girth and Hamiltonian. Moreover, planarity and 1-planarity are also investigated here.

Periodica Mathematica Hungarica, 1976

It is well-knot~n tha~ its automorphism group A(X o H) must contain the regular subgroup L G corresponding to the set of left multiplication~ by elements of G. This paper is concerned with minimizing the index [A(Xo, t/):L a] for given G, in particular when this index is always greater than 1. If G is a.beli~n but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

Rocky Mountain Journal of Mathematics, 2018

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.

Illinois Journal of Mathematics, 2010

We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphism generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually CAT(0); it has a finite index subgroup which acts geometrically on a right-angled building.

Proceedings of the sixth annual ACM …, 1974

The isomorphism problem for graphs G 1 and G 2 is to determine if there exists a oneto-one mapping of the vertices of G 1 onto the vertices of G 2 such that two vertices of G 1 are adjacent if and only if their images in G 2 are adjacent.

Bulletin of The Iranian Mathematical Society, 2014

On the planarity of a graph related to the join of subgroups of a finite group .