Irreversible $k$-Threshold Conversion Number of Circulant Graphs

Abstract

An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k -threshold conversion process on a graph G = V , E is an iterative process which begins by choosing a set S 0 ⊆ V , and for each step t t = 1 , 2 , ⋯ , , S t is obtained from S t − 1 by adjoining all vertices that have at least k neighbors in S t − 1 . S 0 is called the seed set of the k -threshold conversion process, and if S t = V G for some t ≥ 0 , then S 0 is an irreversible k -threshold conversion set (IkCS) of G . The k -threshold conversion number of G (denoted by ( C k G ) is the minimum cardinality of all the IkCSs of G . In this paper, we determine C 2 G for the circulant graph C n 1 , r when r is arbitrary; we also find C 3 C n 1 , r when r = 2 , 3 . We also introduce an upper bound for C 3 C n 1 , 4 . Finally, we suggest an upper bound for C 3 C n 1 , r if n ≥ 2 r + 1 and n ≡ 0 mod 2 r + 1 .

معامل جديد متعلق بعمليات التحول غير العكوس ذو عتبة الانتشار k في البيان

“…As an immediate consequence of the definition of IkCS, when studying an Irreversible kthreshold conversion process on a graph 𝐺 = (𝑉, 𝐸) all vertices {𝑣 ∈ 𝑉 ; 𝑑𝑒𝑔 (𝑣) < 𝑘} must be included in the seed set 𝑆 0 , otherwise the process will fail because none of these vertices can satisfy the conversion rule. These vertices are called k-immune vertices 7 .…”

“…However, it would also leave 𝐵 3 , 𝐵 4 fully unconverted, this means both (1,8), (3,8) need to be included in 𝑆 0 to avoid having unconverted 𝑊 7 , 𝑋 7 , 𝑌 7 , 𝑍 7 .Therefore, 5 vertices from the 5 columns 𝐶𝑂 𝑗 : 4 ≤ 𝑗 ≤ 8 must be included in 𝑆 0 or else the process automatically fails. Case 2.c: In order to prevent leaving any of 𝐵 1 , 𝐵 2 , 𝐵 3 , 𝐵 4 entirely unconverted, the two chosen converted vertices should be (2,4), (2,7). However, that would leave 𝑊 6 , 𝑋 6 , 𝑌 6 , 𝑍 6 unconverted and the process automatically fails.…”

Irreversible k-threshold conversion number of some graphs

“…In Ref. [7] Shaheen et al studied irreversible k-threshold conversion processes on circulant graphs. In Ref.…”