## Missouri Journal of Mathematical Sciences

- Missouri J. Math. Sci.
- Volume 30, Issue 1 (2018), 85-92.

### The Smallest Self-dual Embeddable Graphs in a Pseudosurface

Ethan Rarity, Steven Schluchter, and J. Z. Schroeder

#### Abstract

A proper embedding of a graph $G$ in a pseudosurface $P$ is an embedding in which the regions of the complement of $G$ in $P$ are homeomorphic to discs and a vertex of $G$ appears at each pinchpoint of $P$; we say that a proper embedding of $G$ in $P$ is self dual if there exists an isomorphism from $G$ to its topological dual. We determine five possible graphs with 7 vertices and 13 edges that could be self-dual embeddable in the pinched sphere, and we establish, by way of computer-powered methods, that such a self-embedding exists for exactly two of these five graphs.

#### Article information

**Source**

Missouri J. Math. Sci., Volume 30, Issue 1 (2018), 85-92.

**Dates**

First available in Project Euclid: 16 August 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.mjms/1534384958

**Mathematical Reviews number (MathSciNet)**

MR3844394

**Subjects**

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

**Keywords**

graph embedding self-dual embedding surgeries pseudosurface

#### Citation

Rarity, Ethan; Schluchter, Steven; Schroeder, J. Z. The Smallest Self-dual Embeddable Graphs in a Pseudosurface. Missouri J. Math. Sci. 30 (2018), no. 1, 85--92. https://projecteuclid.org/euclid.mjms/1534384958