Domination problems with no conflicts

• Published in: Discrete Applied Mathematics Vol. 244; pp. 78 - 88
• Main Authors: Cornet, Alexis, Laforest, Christian
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 31.07.2018; Elsevier BV; Elsevier
• Subjects: Automata theory; Combinatorics; Complexity theory; Computational Complexity; Computer Science; Conflicts; Design optimization; Dominating set; Forbidden pairs; Graph theory; Independent dominating set; NP-completeness; Operations Research; Polynomials; Forbidden pairs; Independent dominating set; Graph theory; Conflicts; Dominating set; NP-completeness
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2018.03.018

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.