C = Cycle modular sequence E = Edges set/matrix F = Former main breakpoints set/matrix u , v = Vertices in a graph p , q , r , s = Indices in a sequence b ( E , u , v ) ∈ { 0 , 1 } = { 0 if ( u , v ) is an edge in E 1 if ( u , v ) is not an edge in E f ( E , F , u , v ) ∈ { 0 , 1 } = { 0 if b ( E , u , v ) ≠ 1 ∨ b ( F , u , v ) ≠ 0 1 if b ( E , u , v ) = 1 ∧ b ( F , u , v ) = 0 c ( C , E , p ∈ C , q ∈ C , r ∈ C , s ∈ C ) ∈ { 0 , 1 , 2 , 3 } = b ( E , C p − 1 , C q + 1 ) + b ( E , C q , C s ) + b ( E , C p , C r ) d ( C , E , F , p ∈ C , q ∈ C , r ∈ C , s ∈ C ) ∈ { 0 , 1 , 2 , 3 } = f ( E , F , C p − 1 , C q + 1 ) + f ( E , F , C q , C s ) + f ( E , F , C p , C r ) b , c , d , f ∈ O ( 1 ) {\displaystyle {\begin{array}{lcl}C&=&{\text{Cycle modular sequence}}\\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}}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}}}
m ( C , F , E ) ∈ { 0 , 1 , 2 , 3 } = min p , q , r , s ∈ C ( p ≤ q ) ∧ ( q < r ) ∧ ( q < s ) ∧ ( | s − r | = 1 ) ( c ( C , E , p , q , r , s ) ≠ 0 ) → ( d ( C , E , F , p , q , r , s ) ≠ 0 ) c ( C , E , p , q , r , s ) {\displaystyle {\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}}}