Leírás
Edge-elimination is an operation of removing an edge together
with its endvertices and suppressing the resulting $2$-valent
vertices. We study the effect of this operation on the cyclic
connectivity of a cubic graph. Disregarding a small number of
cubic graphs with no more than six vertices, this operation
cannot decrease cyclic connectivity by more than two. We show
that apart from three exceptional graphs (the cube, the twisted
cube, and the Petersen graph) every 2-connected cubic graph on
at least eight vertices contains an edge whose elimination
decreases cyclic connectivity by at most one. The proof reveals
an unexpected behaviour of connectivity~$6$, which requires a
detailed structural analysis featuring the Isaacs flower snarks
and their natural generalisation, the twisted Isaacs graphs, as
forced structures. A complete characterisation of this family,
which includes the Heawood graph as a sporadic case, serves as
the main tool for excluding the existence of exceptional graphs
in connectivity $6$. As an application we show that every
cyclically $5$-edge-connected cubic graph has a decycling set
of vertices whose removal leaves a tree and the set itself has
at most one edge between its vertices. This strengthens a
classical result of Payan and Sakarovitch (1975) about the
structure of minimum decycling sets in cyclically
$4$-edge-connected graphs.