Domination problems with no conflicts.

*(English)*Zbl 1387.05181Summary: Domination problems have been studied in graph theory for decades. In most of them, it is NP-complete to find an optimal solution, while it is easy (and even trivial in some cases) to find a solution in polynomial time, regardless of its size.

In recent works, authors added conflicts to classical discrete optimization problems. In this paper, a conflict is a pair of vertices that cannot be both in a solution. Set of conflicts can be viewed as edges of a so called conflict graph. An instance is then a support graph and a conflict graph. With these new constraints, the existence of a solution (dominating set or independent dominating set) with no conflicts is no more guaranteed. We explore this subject and we prove that it is NP-complete to decide the existence of a solution even in very restricted classes of graphs and conflicts (sparse or dense). We also propose polynomial algorithms for some sub-cases, using deterministic finite automata.

In recent works, authors added conflicts to classical discrete optimization problems. In this paper, a conflict is a pair of vertices that cannot be both in a solution. Set of conflicts can be viewed as edges of a so called conflict graph. An instance is then a support graph and a conflict graph. With these new constraints, the existence of a solution (dominating set or independent dominating set) with no conflicts is no more guaranteed. We explore this subject and we prove that it is NP-complete to decide the existence of a solution even in very restricted classes of graphs and conflicts (sparse or dense). We also propose polynomial algorithms for some sub-cases, using deterministic finite automata.

##### MSC:

05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |

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\textit{A. Cornet} and \textit{C. Laforest}, Discrete Appl. Math. 244, 78--88 (2018; Zbl 1387.05181)

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##### References:

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