Examples of Jump-Critical Ordered Sets
Mohamed H. El-Zahar, Ivan Rival · SIAM Journal on Algebraic and Discrete Methods · 1985
For an ordered set P and for a linear extension L of P let $s ( P,L )$ stand for the number of ordered pairs $ ( x,y )$ of elements of P such that x is an immediate successor of y in L but x is not even above y in P. Put $s ( P ) = \min \{ s ( P,L ) |L\,{\text{linear extension of}}\,P \}$, the jump number of P. Call an ordered set Pjump-critical if $s ( P - \{ x \} ) < s ( P )$ for any $x \in P$. There are precisely seventeen jump-critical ordered sets with jump number at most three.