User:Realazthat/Notes/Scrap

Du

${\begin{array}{lcl}C&=&{\text{Cycle modular sequence of vertices}}\\E&=&{\text{Edges set/matrix}}\\F&=&{\text{Former main breakpoints set/matrix}}\\u,v&=&{\text{Vertices in a graph}}\\p,q,r,s&=&{\text{Indices in a sequence}}\\(C_{p},C_{p-1})&=&{\text{Current main breakpoint}}\\\left[p,q\right]\in C&=&{\text{Segment to cut, ranged from }}p{\text{ to }}q{\text{ in }}C\\\left(r,s\right)\in C&=&{\text{Pair of neighboring vertices representing an edge to insert the segment in between}}\\b\left(E,u,v\right)\in \left\{0,1\right\}&=&{\begin{cases}0&{\text{if }}\left(u,v\right){\text{ is an edge in }}E\\1&{\text{if }}\left(u,v\right){\text{ is not an edge in }}E\end{cases}}\\f\left(E,F,u,v\right)\in \left\{0,1\right\}&=&{\begin{cases}0&{\text{if }}b\left(E,u,v\right)\neq 1\vee b\left(F,u,v\right)\neq 0\\1&{\text{if }}b\left(E,u,v\right)=1\wedge b\left(F,u,v\right)=0\end{cases}}\\c\left(C,E,p\in C,q\in C,r\in C,s\in C\right)\in \left\{0,1,2,3\right\}&=&b\left(E,C_{p-1},C_{q+1}\right)+b\left(E,C_{q},C_{s}\right)+b\left(E,C_{p},C_{r}\right)\\d\left(C,E,F,p\in C,q\in C,r\in C,s\in C\right)\in \left\{0,1,2,3\right\}&=&f\left(E,F,C_{p-1},C_{q+1}\right)+f\left(E,F,C_{q},C_{s}\right)+f\left(E,F,C_{p},C_{r}\right)\\b,c,d,f\in O\left(1\right)\\\end{array}}$

${\begin{array}{lcl}m\left(C,F,E\right)\in \left\{0,1,2,3\right\}&=&{\underset {\begin{array}{c}p,q,r,s\in C\\\left(p\leq q\right)\wedge \left(q<r\right)\wedge \left(q<s\right)\wedge \left(\left|s-r\right|=1\right)\\\left(c\left(C,E,p,q,r,s\right)\neq 0\right)\rightarrow \left(d\left(C,E,F,p,q,r,s\right)\neq 0\right)\end{array}}{\min }}c\left(C,E,p,q,r,s\right)\\\end{array}}$