Secure Distance-k Domination Number of Graphs

Abstract

This paper introduces the concept of secure distance-k domination in graphs. Fundamental properties and bounds for the secure distance-k domination number are established, including its relationship with both the distance-k domination number and the classical domination number. We characterize graphs with secure distance-k domination number equal to 1 in terms of their diameter, and provide a complete characterization for when this parameter attains its maximum value. Additionally, the behavior of secure distance-k dominating sets on disconnected graphs is evaluated, establishing an additive property over graph components. Furthermore, we derive a necessary and sufficient neighborhood containment condition for a distance-k dominating set to be secure, which yields a structural characterization of graphs where the distance-k domination number and its secure counterpart coincide. Finally, we determine exact values of the secure distance-k domination number for binary graph operations, specifically the join and corona products, and for the path and cycle graphs.