  We introduce the $k$-bonacci polyominoes, a new family of polyominoes
associated with the binary words avoiding $k$ consecutive $1$'s, also called
generalized $k$-bonacci words. The polyominoes are very entrancing objects,
considered in combinatorics and computer science. The study of polyominoes
generates a rich source of combinatorial ideas. In this paper we study some
properties of $k$-bonacci polyominoes. Specifically, we determine their
recursive structure and, using this structure, we enumerate them according to
their area, semiperimeter, and length of the corresponding words. We also
introduce the $k$-bonacci graphs, then we obtain the generating functions for
the total number of vertices and edges, the distribution of the degrees, and
the total number of $k$-bonacci graphs that have a Hamiltonian cycle.
