Even-hole-free graph explained

In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs.

demonstrated that every even-hole-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques), which settled a conjecture by Reed. The proof was later shown to be flawed by, who gave a correct proof.

Recognition

gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in

{ l O}(n ^{40} ) time.^[1] later improved this to { l O}(n ^{19} ) . and improved this to { l O}(n ^{11} ) time.The best currently known algorithm is given by which runs in { l O}(n ^{9)} time.

While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex.

It is unknown whether graph coloring and the maximum independent set problem can be solved in polynomial time on even-hole-free graphs, or whether they are NP-complete.However the maximum clique can be found in even-hole-free graphs in polynomial time.

Notes and References

1. present their algorithm and assert that it runs in polynomial time without giving an explicit analysis. estimate that it runs in "time about { l O}(n ^{40} ) ."