Irreversible 2-conversion set in graphs of bounded degree

Kynčl, Jan; Lidický, Bernard; Vyskočil, T

doi:10.23638/dmtcs-19-3-5

“…Convexity. Activation problems appear in the literature under a number of different names, such as r-neighbor bootstrap percolation [2-6, 18, 42, 51, 54], dynamic monopolies [15,19,46,55,58], irreversible conversion [17,43,47,56], or graph convexities, and were studied by researchers of various fields. As mentioned in the introduction, in the particular case in which all thresholds are equal to 2, generalized TSS model is also called 2-neighbor bootstrap percolation or P 3 -convexity, which is studied in the field of graph convexities.…”

Section: Preliminariesmentioning

confidence: 99%

Target set selection with maximum activation time

Keiler¹,

Lima²,

Maia³

et al. 2020

Preprint

0

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A target set selection model is a graph G with a threshold function τ : V → N upper-bounded by the vertex degree. For a given model, a set S0 ⊆ V (G) is a target set if V (G) can be partitioned into non-empty subsets S0, S1, . . . , St such that, for i ∈ {1, . . . , t}, Si contains exactly every vertex v having at least τ (v) neighbors in S0 ∪ • • • ∪ Si−1. We say that t is the activation time tτ (S0) of the target set S0. The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set S0 that maximizes tτ (S0). That is, given a graph G, a threshold function τ in G, and an integer k, the objective of the TSS-time problem is to decide whether G contains a target set S0 such that tτ (S0) ≥ k. Let τ * = max v∈V (G) τ (v). Our main result is the following dichotomy about the complexity of TSS-time when G belongs to a minor-closed graph class C: if C has bounded local treewidth, the problem is FPT parameterized by k and τ ; otherwise, it is NP-complete even for fixed k = 4 and τ = 2. We also prove that, with τ * = 2, the problem is NP-hard in bipartite graphs for fixed k = 5, and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set S0 in a given tree maximizing tτ (S0). ACM Subject ClassificationMathematics of computing → Graph algorithms.

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“…[8] Shaheen et al determined C 2 ðGÞ and C 3 ðGÞ for the strong grid graphs P m @P n when m ¼ 2; 3. For further information on the irreversible k-threshold conversion problem on graphs see Centeno et al [9], Takaoka and Ueno [10], Kyn cl et al [11]. A generalized Jahangir graph J s;m for m ≥ 2 is a graph on sm þ 1vertices, i.e.…”

Section: Introductionmentioning

confidence: 99%

Irreversible k-threshold conversion number of some graphs

Shaheen

¹

,

Mahfud²,

Kassem³

2022

AJMS

0

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PurposeThis paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule is determined at the beginning of the study. These processes can be modeled into graph theoretical models where the vertex set V(G) represents the set of individuals on which the conversion is spreading.Design/methodology/approachThe irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process which starts 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 is called an irreversible k-threshold conversion set (IkCS) of G if S_t = V(G) for some t = 0. The minimum cardinality of all the IkCSs of G is referred to as the irreversible k-threshold conversion number of G and is denoted by C_k (G).FindingsIn this paper the authors determine C_k (G) for generalized Jahangir graph J_(s,m) for 1 < k = m and s, m are arbitraries. The authors also determine C_k (G) for strong grids P_2? P_n when k = 4, 5. Finally, the authors determine C_2 (G) for P_n? P_n when n is arbitrary.Originality/valueThis work is 100% original and has important use in real life problems like Anti-Bioterrorism.

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Irreversible $k$ -Threshold Conversion Number of Circulant Graphs

Shaheen

¹

,

Mahfud

²

,

Kassem

³

2022

Journal of Applied Mathematics

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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 .

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Irreversible 2-conversion set in graphs of bounded degree

Cited by 3 publications

References 0 publications

Target set selection with maximum activation time

Target set selection with maximum activation time

Irreversible k-threshold conversion number of some graphs

Irreversible $k$ -Threshold Conversion Number of Circulant Graphs

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Irreversible 2-conversion set in graphs of bounded degree

Cited by 3 publications

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Target set selection with maximum activation time

Target set selection with maximum activation time

Irreversible k-threshold conversion number of some graphs

Irreversible k -Threshold Conversion Number of Circulant Graphs

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Irreversible $k$ -Threshold Conversion Number of Circulant Graphs