▁gives ▁the ▁probability ▁that ▁every ▁pair ▁of ▁vertices ▁in ▁G ▁remains ▁connected ▁after ▁the ▁edges ▁fail . ▁The ▁connection ▁to ▁the ▁T ut te ▁polynomial ▁is ▁given ▁by ▁ ▁D ich rom atic ▁polynomial ▁T ut te ▁also ▁defined ▁a ▁closer ▁ 2 - variable ▁general ization ▁of ▁the ▁chrom atic ▁polynomial , ▁the ▁dich rom atic ▁polynomial ▁of ▁a ▁graph . ▁ ▁This ▁is ▁ ▁where ▁ ▁is ▁the ▁number ▁of ▁connected ▁components ▁of ▁the ▁sp anning ▁sub graph ▁( V , A ). ▁ ▁This ▁is ▁related ▁to ▁the ▁cor ank - null ity ▁polynomial ▁by ▁ ▁The ▁dich rom atic ▁polynomial ▁does ▁not ▁general ize ▁to ▁mat roid s ▁because ▁k ( A ) ▁is ▁not ▁a ▁mat roid ▁property : ▁different ▁graphs ▁with ▁the ▁same ▁mat roid ▁can ▁have ▁different ▁numbers ▁of ▁connected ▁components . ▁ ▁Rel ated ▁polynomials ▁ ▁Martin ▁polynomial ▁ ▁The ▁Martin ▁polynomial ▁ ▁of ▁an ▁orient ed ▁ 4 - regular ▁graph ▁ ▁was ▁defined ▁by ▁Pierre ▁Martin ▁in ▁ 1 9 7 7 . ▁He ▁showed ▁that ▁if ▁G ▁is ▁a ▁plane ▁graph ▁and ▁ ▁is ▁its ▁directed ▁med ial ▁graph , ▁then ▁ ▁Alg orithms ▁ ▁De let ion – cont raction ▁ ▁The ▁delet ion – cont raction ▁rec urrence ▁for ▁the ▁T ut te ▁polynomial , ▁▁ ▁immediately ▁yields ▁a ▁recursive ▁algorithm ▁for ▁computing ▁it : ▁choose ▁any ▁such ▁edge ▁e ▁and ▁repeatedly ▁apply ▁the ▁formula ▁until ▁all ▁edges ▁are ▁either ▁loops ▁or ▁brid ges ; ▁the ▁resulting ▁base ▁cases ▁at