Finding Short Cycles in an Embedded Graph in Polynomial Time

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Ren, Han
Cao, Ni
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Abstract
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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]
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Mathematics - Combinatorics, 05C10, 05C30
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