On the Maximum Number of Vertices of Critically Embedded Graphs

TL;DR

This paper investigates the maximum number of vertices in planar critical graphs embedded in the Euclidean plane with boundary points, under a geometric condition related to edge vectors summing to zero at non-boundary vertices.

Contribution

It establishes a sharp upper bound for the number of vertices in such critical graphs with degrees 3 or 4, advancing understanding of their combinatorial structure.

Findings

Sharp upper bound for vertices in planar critical graphs

Maximum vertices achieved with degree constraints

Characterization of extremal critical graphs

Abstract

Define a boundary point of a graph which is embedded in the Euclidean plane a vertex which is incident to only one edge. In this paper we consider graphs which are embedded in the Euclidean plane with a finite number of boundary points. The simple geometric condition we impose on them is that the sum of unit vectors of edges extending from each non-boundary vertex will be equal to zero. We call such a graph a critical graph and ask to maximise the number of vertices of critical graphs with a given size of boundary. The main results of this paper give a sharp upper bound for the maximum number of vertices of planar critical graphs, where the degree of each non-boundary vertex is 3 or 4.