isolate pendant domination number in semi-total point graphs and its application

Abstract

Let G be a non-trivial graph. A dominating set I ⊂ V (G) is called an Isolate Pendant Dominating set (IPD-set) if the induced subgraph 〈I〉 has maximum degree at most one and contains at least one isolate vertex and at least one pendant vertex. The minimum cardinality of an IPD-set is called the IP domination number and is denoted by γ01(G). In this paper, we study the IP domination numberbfor semi-total point graphs. We determine exact values of γ_{01}(T_2(G)) for several well-known graph classes, including paths, cycles, and complete graphs. Furthermore, we provide a characterization of IPD-sets in the join of two graphs and examine the behavior of γ01 under the join operation for both arbitrary graphs and semi-total point graphs. Finally, we obtain lower and upper bounds for γ_{01}(T_2(G)) in terms of the order and size of G. The practical relevance of the proposed concepts is illustrated through an application in pest control management.