A characterization of graphs with at most four boundary vertices

TL;DR

This paper characterizes graphs with up to four boundary vertices using specific families of graphs, extending previous work on graph boundaries and their properties.

Contribution

It provides a complete characterization of graphs with three and four boundary vertices, identifying specific families that describe their structure.

Findings

Graphs with three boundary vertices are characterized by two infinite families.

Graphs with four boundary vertices are characterized by eight families, five of which are infinite.

The work extends earlier boundary concepts and their isoperimetric inequalities.

Abstract

Steinerberger defined a notion of boundary for a graph and established a corresponding isoperimetric inquality. Hence, "large" graphs have more boundary vertices. In this paper, we first characterize graphs with three boundary vertices in terms of two infinite families of graphs. We then completely characterize graphs with four boundary vertices in terms of eight families of graphs, five of which are infinite. This parallels earlier work by Hasegawa and Saito as well as M\"uller, P\'or, and Sereni on another notion of boundary defined by Chartrand, Erwin, Johns, and Zhang.