Critically indecomposable graphs Original Research Article

An undirected graph image with a specific subset image is called image-critical if image and image, induced subgraph on image, are indecomposable but image is decomposable for every image. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191–205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the image-property, Theoretical Computer Science 132 (1994) 151–178], who deal with the case where image is empty.