Complexity results about locally bipartite graphs 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.
Locally bipartite graphs obviously contain the locally independent (= triangle-free) graphs. According to graphclasses.org, most of the standard graph problems are already NP-complete for triangle-free graphs, and therefore also NP-complete for locally bipartite graphs. The two exceptions are clique (which is obviously polynomial for locally bipartite graphs because the maximum clique is a triangle) and clique cover, which is polynomial for triangle-free graphs but might be harder for locally bipartite graphs.