Total Domination, Connected Vertex Cover and Steiner Tree with Conflicts

• Published in: Discrete Mathematics and Theoretical Computer Science Vol. 19 no. 3; no. Graph Theory; p. 1
• Main Authors: Cornet, Alexis, Laforest, Christian
• Format: Journal Article
• Language: English
• Published: DMTCS 20.12.2017; Discrete Mathematics & Theoretical Computer Science
• Subjects: [info.info-ro] computer science [cs]/operations research [cs.ro]; Computer Science; conflicts; connected vertex cover; forbidden pairs; graph theory; graphs; Mathematical optimization; Mathematical research; np- completeness; Operations Research; steiner tree; total dominating set; Trees (Graph theory); France; forbidden pairs; graph theory; Steiner tree; Graphs; conflicts; connected vertex cover; total dominating set; NP- completeness
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.23638/DMTCS-19-3-17

Total dominating set, connected vertex cover and Steiner tree are well-known graph problems. Despite the fact that they are NP-complete to optimize, it is easy (even trivial) to find solutions, regardless of their size. In this paper, we study a variant of these problems by adding conflicts, that are pairs of vertices that cannot be both in a solution. This new constraint leads to situations where it is NP-complete to decide if there exists a solution avoiding conflicts. This paper proposes NP-completeness proofs of the existence of a solution for different restricted classes of graphs and conflicts, improving recent results. We also propose polynomial time constructions in several restricted cases and we introduce a new parameter, the stretch, to capture the locality of the conflicts.