Explicit formulas for graphical Stirling and Bell numbers are known for relatively few graph families. We derive exact expressions for three classes whose independence structure admits a complete combinatorial description: complete multipartite graphs, the graph obtained from a balanced complete bipartite graph by deleting a perfect matching, and the Mycielskian of a star. For complete multipartite graphs we express the graphical Stirling number as a convolution of classical Stirling numbers across the partite classes, and we recover the known factorization of the graphical Bell number as a product of classical Bell numbers. For the matching-deleted graph we show that its graphical Bell number is a binomial convolution of squared Bell numbers, which we identify as a moment of a product of two independent Poisson random variables with unit mean. This representation yields log-convexity of the sequence, a sharp exponential lower bound, a two-sided estimate, and a Laplace-transform identity. For the Mycielskian of a star, a decomposition according to the block containing the original center vertex, together with Vandermonde’s convolution and a Stirling recurrence, gives a single-sum closed form for the graphical Stirling numbers, from which two explicit evaluations follow. Several resulting integer sequences appear in the OEIS, and one Bell-number sequence appears not to be currently recorded there.