A graph is locally bipartite if the open neighborhood of every vertex induces a bipartite graph. (According to searches the same name might be used for something else related to surfaces).
Which NP-hard for general graphs problems become polynomial for locally bipartite graphs and which remain NP-hard?
Especially interested in clique and coloring.
Are there inclusions between locally bipartite and other graph classes?
Added According to a paper they are also called "almost bipartite" and their complements are generalized line graphs which are claw free.