  Motivated by the concept of well-covered graphs, we define a graph to be
well-bicovered if every vertex-maximal bipartite subgraph has the same order
(which we call the bipartite number). We first give examples of them, compare
them with well-covered graphs, and characterize those with small or large
bipartite number. We then consider graph operations including the union, join,
and lexicographic and cartesian products. Thereafter we consider simplicial
vertices and 3-colored graphs where every vertex is in triangle, and conclude
by characterizing the maximal outerplanar graphs that are well-bicovered.
