Triangle-free projective-planar graphs with diameter two: domination and characterization

TL;DR

This paper characterizes all triangle-free projective-planar graphs with diameter two and explores applications including calculating clique number analogues for complex colored mixed graphs.

Contribution

It provides a complete characterization of triangle-free projective-planar graphs with diameter two, extending Plesník's planar graph results to the projective plane.

Findings

Characterization of all such graphs with diameter two

Application to colored mixed graphs and clique number analogues

Extension of planar graph diameter results to projective-planar graphs

Abstract

In 1975, Plesn\'ik characterized all triangle-free planar graphs as having a diameter $2$. We characterize all triangle-free projective-planar graphs having a diameter $2$ and discuss some applications. In particular, the main result is applied to calculate the analogue of clique numbers for graphs, namely, colored mixed graphs, having different types of arcs and edges.