Minimal Non-simple and Minimal Non-cosimple Sets in Binary Images on Cell Complexes T. Yung Kong Department of Computer Science, Queens College, City University of New York, 65-30 Kissena Boulevard, Flushing, NY 11367, U.S.A. Abstract. The concepts of weak component and simple 1 are general- izations, to binary images on the n-cells of n-dimensional cell complexes, of the standard concepts of “26-component” and “26-simple” 1 in bi- nary images on the 3-cells of a 3D cubical complex; the concepts of strong component and cosimple 1 are generalizations of the concepts of “6-component” and “6-simple” 1. Over the past 20 years, the problems of determining just which sets of 1’s can be minimal non-simple, just which sets can be minimal non-cosimple, and just which sets can be minimal non-simple (minimal non-cosimple) without being a weak (strong) fore- ground component have been solved for the 2D cubical and hexagonal, 3D cubical and face-centered-cubical, and 4D cubical complexes. This paper solves these problems in much greater generality, for a very large class of cell complexes of dimension ≤ 4. 1 Introduction In a binary image, the n-dimensional cells of an n-dimensional cell complex (most often, the 2D or 3D cubical complex) are labeled 1 or 0. Cells labeled 1 are referred to as 1’s of the image, and cells labeled 0 are referred to as 0’s. We say that a 1 of the image is simple if “the topology of image is preserved” (in a sense which will be made precise in Sect. 4) when that 1 is changed into a 0. We say that a 1 is cosimple if the topology of the image is preserved in another, complementary, sense when the 1 is changed into a 0. In the case of the 2D cubical complex, these are two of the oldest con- cepts of digital topology, and date back to the 1960’s. Rosenfeld’s concept of an “8-deletable” pixel in [20] is mathematically equivalent to our concept of a simple 1 in a binary image on the 2D cubical complex. The concept of a “4-deletable” pixel in [20] is similarly equivalent to our concept of a cosimple 1. Today, simple and cosimple 1’s in binary images on the 2D cubical complex are often called “8-simple” and “4-simple”, respectively. In binary images on the 3D cubical complex, simple 1’s are often called “26-simple”, cosimple 1’s are often called “6-simple”, and a number of non-trivial characterizations of such 1’s have been published (e.g., in [2,21]). A subset of the set of 1’s of a binary image is said to be simple (cosimple) if the elements of that subset can be arranged in a sequence D 1 ,...,D k in which each element D i is simple (cosimple) after its predecessors D 1 ,...,D i-1 have all A. Kuba, L.G. Ny´ ul, and K. Pal´agyi (Eds.): DGCI 2006, LNCS 4245, pp. 169–188, 2006. c  Springer-Verlag Berlin Heidelberg 2006