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Article Dans Une Revue Discrete Mathematics and Theoretical Computer Science Année : 2022

Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree

Résumé

We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.

Dates et versions

hal-04152217 , version 1 (05-07-2023)

Identifiants

Citer

Julien Baste, Stefan Ehard, Dieter Rautenbach. Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree. Discrete Mathematics and Theoretical Computer Science, 2022, 24 (1), ⟨10.46298/dmtcs.6844⟩. ⟨hal-04152217⟩
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