We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and compare their performance on various kinds of graphs. We demonstrate that the first performs better in most cases, but the second performs better for very sparse graphs. Also included is a short proof that the upper domination of any generalized Petersen graph P (n, k) is equal to n.

The secure domination problem, a variation of the domination problem with some important real-world applications, is considered. Very few algorithmic attempts to solve this problem have been presented in literature, and the most successful to date is a binary programming formulation which is solved using CPLEX. A new binary programming formulation is proposed here which requires fewer constraints and fewer binary variables than the existing formulation. It is implemented in CPLEX, and tested on certain families of graphs that have previously been considered in the context of secure domination. It is shown that the runtime required for the new formulation to solve the instances is significantly less than that of the existing formulation. An extension of our formulation that solves the related, but further constrained, secure connected domination problem is also given; to the best of the authors' knowledge, this is the first such formulation in literature.

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n, k) is equal to n.

We consider the flower snarks, a widely studied infinite family of 3-regular graphs. For the Flower snark J n on 4n vertices, it is trivial to show that the domination number of J n is equal to n. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure domination and weak Roman domination numbers for flower snarks.

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