Exponential Domination Critical and Stability in Some Graphs

Abstract

An exponential dominating set of graph [Formula: see text] is a kind of distance domination subset [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is the length of a shortest path in [Formula: see text] if such a path exists, and [Formula: see text] otherwise. The minimum exponential domination number, [Formula: see text] is the smallest cardinality of an exponential dominating set. The minimum exponential domination number, [Formula: see text] can be decreased or increased by removal of some vertices from [Formula: see text]. In this paper, we investigate of this phenomenon which is referred to critical and stability in graphs.

Exponential independence

“…Now, let S be a maximum exponential independent set of T . If a ∈ S, then S is an exponential independent set of T − a = T 1 (k), which contradicts α e (T 1 (k)) = k. Hence, a ∈ S, which implies x 1 ∈ S. For k ∈ [2], it follows easily that S is as stated in (ii). Now, let k ≥ 3.…”

Paired domination stability in graphs

“…Stability for other domination type parameters has been studied in the literature. For example, total domination stability, 2-rainbow domination stability, exponential domination stability, Roman domination stability are studied in [1,2,12,15,16], among other papers. In this paper we study the paired version of domination stability.…”

Disjunctive total domination stability in graphs

“…The total domination stability is defined as the minimum size of a non‐isolating set of vertices in G whose removal changes (increases, ‐stability of G , or decreases, ‐stability of G ) the total domination number. Moreover, the stability of some other domination parameters such as 2‐rainbow domination, 19 weak domination, 20 strong domination, 21 and exponential domination 22 is introduced. In this paper, we introduce the same concept for disjunctive total domination.…”