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Monotone Decontamination of Arbitrary Dynamic Graphs with Mobile Agents

Rajashree Bar, Daibik Barik, Adri Bhattacharya, Partha Sarathi Mandal

TL;DR

This work addresses monotone decontamination on arbitrary dynamic graphs under $1$-interval connectivity, introducing two adversarial edge-dynamics models: finite-time edge appearance ($T$ bound) and indefinite edge disappearance. It derives lower bounds that scale with the graph diameter $d$, size $n$, and cyclomatic number $k=\mu(G)$, showing, for example, $d$ or $n-1$ bounds in FTEA and $\max\{d,\,k+1\}$ or $n-1$ in IDED depending on $k$ relative to $n$. The paper then contributes three DFS/BFS-based algorithms: Uni-Decontamination for $k\ge n$ using $n$ agents, Modified-Decontamination for $k<n$ using $d+k$ agents, and Infinite-Decontamination for IDED using $d+2k$ agents, each ensuring monotonicity despite edge disappearances. Together, these results establish foundational bounds and constructive strategies for monotone network decontamination in dynamic environments and lay groundwork for tighter future analyses.

Abstract

Network decontamination is a well-known problem, in which the aim of the mobile agents should be to decontaminate the network (i.e., both nodes and edges). This problem comes with an added constraint, i.e., of \emph{monotonicity}, in which whenever a node or an edge is decontaminated, it must not get recontaminated. Hence, the name comes \emph{monotone decontamination}. This problem has been relatively explored in static graphs, but nothing is known yet in dynamic graphs. We, in this paper, study the \emph{monotone decontamination} problem in arbitrary dynamic graphs. We designed two models of dynamicity, based on the time within which a disappeared edge must reappear. In each of these two models, we proposed lower bounds as well as upper bounds on the number of agents, required to fully decontaminate the underlying dynamic graph, monotonically. Our results also highlight the difficulties faced due to the sudden disappearance or reappearance of edges. Our aim in this paper has been to primarily optimize the number of agents required to solve monotone decontamination in these dynamic networks.

Monotone Decontamination of Arbitrary Dynamic Graphs with Mobile Agents

TL;DR

This work addresses monotone decontamination on arbitrary dynamic graphs under -interval connectivity, introducing two adversarial edge-dynamics models: finite-time edge appearance ( bound) and indefinite edge disappearance. It derives lower bounds that scale with the graph diameter , size , and cyclomatic number , showing, for example, or bounds in FTEA and or in IDED depending on relative to . The paper then contributes three DFS/BFS-based algorithms: Uni-Decontamination for using agents, Modified-Decontamination for using agents, and Infinite-Decontamination for IDED using agents, each ensuring monotonicity despite edge disappearances. Together, these results establish foundational bounds and constructive strategies for monotone network decontamination in dynamic environments and lay groundwork for tighter future analyses.

Abstract

Network decontamination is a well-known problem, in which the aim of the mobile agents should be to decontaminate the network (i.e., both nodes and edges). This problem comes with an added constraint, i.e., of \emph{monotonicity}, in which whenever a node or an edge is decontaminated, it must not get recontaminated. Hence, the name comes \emph{monotone decontamination}. This problem has been relatively explored in static graphs, but nothing is known yet in dynamic graphs. We, in this paper, study the \emph{monotone decontamination} problem in arbitrary dynamic graphs. We designed two models of dynamicity, based on the time within which a disappeared edge must reappear. In each of these two models, we proposed lower bounds as well as upper bounds on the number of agents, required to fully decontaminate the underlying dynamic graph, monotonically. Our results also highlight the difficulties faced due to the sudden disappearance or reappearance of edges. Our aim in this paper has been to primarily optimize the number of agents required to solve monotone decontamination in these dynamic networks.

Paper Structure

This paper contains 9 sections, 9 theorems, 1 equation, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Lower Bound
  4. Lower Bound for Finite Time Edge Reappearance Model
  5. Lower Bound for Indefinite Edge Disappearance Model
  6. Algorithm for Network Decontamination
  7. Finite Edge Reappearance Model
  8. Case ($k\ge n$):
  9. Case ($k<n$):

Key Result

theorem thmcountertheorem

There exists a graph $G$ with $n$ nodes, and maximum degree $\frac{n}{2}$, on which no deterministic algorithm can solve network decontamination monotonically with $n-2$ initially co-located agents, in the finite edge reappearance model of dynamicity, where $n>4$.

Figures (3)

  • Figure 1: The complete bipartite graph $K_{4,4}$ with partitions $\{a_1,\dots,a_4\}$ and $\{b_1,\dots,b_4\}$.
  • Figure 2: The wheel graph $G$ on 9 vertices with cyclomatic number $\mu(G)=8$, where the dotted edges imply the disappeared edges.
  • Figure 3: This shows a graph $G$ where $d$ is the diameter and a possible $k=1$. The dotted edge between $v_4$ and $v_5$ denotes the possibility for $k=1$.

Theorems & Definitions (25)

  • definition thmcounterdefinition: Contamination Degree
  • definition thmcounterdefinition: Cyclomatic Number
  • definition thmcounterdefinition: Feedback Edgeskudelic2022feedback
  • definition thmcounterdefinition: Separator Vertex
  • definition thmcounterdefinition: Problem Definition
  • remark thmcounterremark
  • theorem thmcountertheorem: Lower Bound
  • proof
  • theorem thmcountertheorem
  • proof
  • ...and 15 more