An improved binary programming formulation for the secure domination problem

Abstract

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.

Variants of the Domination Number for Flower Snarks

“…and γr(J 8 ) = 13, and the formulation from [5] to confirm the corresponding results for secure domination. Then, suppose there is a value k ≥ 9 such that Theorem 8.6 is true for n = 3, .…”

“…Proof. We use the first formulation for upper domination from [6] to confirm that Γ (J 3 ) = 5. Then, suppose that S is a minimal dominating set for Jn, for n ≥ 4.…”

The Secure Domination Number of Cartesian Products of Small Graphs with Paths and Cycles

“…For small cases, we are able to use the exact formulation from [2], and indeed, we have done so to show that equality holds for n ≤ 16. However, it could be the case that there is some value k ≥ 17 such that equality holds for n = 2, .…”

“…Algorithms exist to solve the secure domination problem, including linear-time algorithms for trees [5] and block graphs [17], as well as algorithms for general graphs [2,3,4]. However, there are very few infinite families of graphs for which the secure domination number is known.…”

“…Hence, it will be necessary to establish the secure domination number for a certain number (depending on the proof) of small cases. Rather than present these small cases here, we will rely on the exact formulation given in Burdett and Haythorpe [2] to handle these.…”