Finding a shortest even hole in polynomial time

An even (respectively, odd) hole in a graph is an induced cycle with length that at least four. Bienstock proved detecting containing given vertex NP-complete. Conforti, Cornuéjols, Kapoor, and Vuškovi? gave the first known polynomial-time algorithm to determine whether contains holes. Chudnovsky, Kawarabayashi, Seymour estimated Conforti et al.'s runs O ( n 40 ) time on n-vertex reduced required 31 . Subsequently, da Silva Vuškovi?, Chang Lu, Lai, Thorup improved 19 , 11 9 respectively. The tractability of determining odd holes has been open for decades until Scott, Seymour, Spirkl time, which Lai al. also 8 By extending Chudnovsky techniques holes, (respectively) ensured finding long shortest) hole. They NP-hardness longest hole, whose reduction works Recently, Cook intriguing missing piece shortest left 16 years by, example, Johnson. We resolve this problem by augmenting even-hole detection into algorithm, running