Finding short cycles in embedded graph in polynomial time

Han Ren,Ni Cao

doi:10.1007/s11464-010-0003-y

Abstract

Let be the set of fundamental cycles of breadth-first-search trees in a graph G, and let be the set of the sums of two cycles in . Then we show the following: (1) contains a shortest Π-twosided cycle in a Π-embedded graph G. This implies the existence of a polynomially bounded algorithm to find a shortest Π-twosided cycle in an embedded graph and thus solves an open problem of Mohar and Thomassen [Graphs on Surfaces, 2001, p. 112]. (2) contains all the possible shortest even cycles in a graph G. Therefore, there are at most polynomially many shortest even cycles in any graph. (3) Let be the set of all the shortest cycles of a graph G. Then is a subset of . Furthermore, many types of shortest cycles are contained in . Infinitely many examples show that there are exponentially many shortest odd cycles, shortest Π-onesided cycles and shortest Π-twosided cycles in some (embedded) graphs.

Highlights

Let C1 be the set of fundamental cycles of breadth-first-search trees in a graph G and C2 the set of the sums of two cycles in C1
C.Thomassen showed that if cycles in a set of cycles satisfy the 3-pathcondition, there exists a polynomial time algorithm that finds a shortest cycle in this set[3]
He showed that the following types of shortest cycles may be found in polynomial time

Summary

Introduction

C.Thomassen showed that if cycles in a set of cycles satisfy the 3-pathcondition, there exists a polynomial time algorithm that finds a shortest cycle in this set[3]. Finding Short Cycles in an Embedded Graph in Polynomial Time 1 Let C1 be the set of fundamental cycles of breadth-first-search trees in a graph G and C2 the set of the sums of two cycles in C1. This implies the existence of a polynomially bounded algorithm to find a shortest Π−twosided cycle in an embedded graph and solves an open problem of B.Mohar and C.Thomassen[2,pp112] Key Words Π−twosided cycle, breadth-first-search tree, embedded graph.

Results

Conclusion