Deciding whether there are infinitely many prime graphs with forbidden   induced subgraphs
  A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set.   We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs. 
Robert Brignall, Hojin Choi, Jisu Jeong, Sang-il Oum
cs.DM math.CO