Finding Short Cycles in an Embedded Graph in Polynomial Time

TL;DR

This paper presents a polynomial-time algorithm for finding shortest cycles in embedded graphs, addressing an open problem by characterizing cycles through fundamental and combined cycle sets.

Contribution

It introduces a novel method to identify shortest cycles in embedded graphs using fundamental cycles and their sums, solving an open problem in graph theory.

Findings

Contains all shortest even cycles in the set C

Contains all shortest odd cycles if the shortest cycle is odd

Provides a polynomial-time algorithm for finding shortest cycles in embedded graphs

Abstract

Let ${\cal{C}}_1$ be the set of fundamental cycles of breadth-first-search trees in a graph $G$ and ${\cal{C}}_2$ the set of the sums of two cycles in ${\cal{C}}_1$. Then we show that $(1) {\cal{C}}={\cal{C}}_1\bigcup{\cal{C}}_2$ contains a shortest $\Pi$-twosided cycle in a $\Pi$-embedded graph $G$;$(2)$ $\cal{C}$ contains all the possible shortest even cycles in a graph $G$;$(3)$ If a shortest cycle in a graph $G$ is an odd cycle, then $\cal{C}$ contains all the shortest odd cycles in $G$. This implies the existence of a polynomially bounded algorithm to find a shortest $\Pi-$twosided cycle in an embedded graph and thus solves an open problem of B.Mohar and C.Thomassen[2,pp112]