Total Outer-Independent Domination Number: Bounds and Algorithms

Abstract

In graph theory, the study of domination sets has garnered significant interest due to its applications in network design and analysis. Consider a graph G(V,E); a subset of its vertices is a total dominating set (TDS) if, for each x∈V(G), there exists an edge in E(G) connecting x to at least one vertex within this subset. If the subgraph induced by the vertices outside the TDS has no edges, the set is called a total outer-independent dominating set (TOIDS). The total outer-independent domination number, denoted as γtoi(G), represents the smallest cardinality of such a set. Deciding if a given graph has a TOIDS with at most r vertices is an NP-complete problem. This study introduces new lower and upper bounds for γtoi(G) and presents an exact solution approach using integer linear programming (ILP). Additionally, we develop a heuristic and a procedure to efficiently obtain minimal TOIDS.

Keywords

• graph theory
• total outer-independent dominating set
• integer linear program
• heuristic algorithm
• time complexity