Characterizations of Some Parity Signed Graphs

TL;DR

This paper explores parity labelings in signed graphs, characterizing balanced graphs and specific structures, and determining the RNA number for certain classes, contributing to understanding graph cuts with nearly equal sides.

Contribution

It provides structural characterizations of various parity signed graphs and computes the RNA number for select graph classes, advancing the theory of graph cuts.

Findings

Characterization of balanced parity signed graphs

Structural descriptions of parity signed stars, bistars, cycles, paths, and bipartite graphs

RNA number calculations for specific graph classes

Abstract

We describe parity labelings of signed graphs; equivalently, cuts of the underlying graph that have nearly equal sides. We characterize the balanced signed graphs which are parity signed graphs. We give structural characterizations of all parity signed stars, bistars, cycles, paths and complete bipartite graphs. The rna number of a graph is the smallest cut size that has nearly equal sides; we find it for a few classes of parity signed graphs.