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Monotony Properties of Connected Visible Graph Searching

Pierre Fraigniaud 1 Nicolas Nisse 2
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Search games are attractive for their correspondence with classical width parameters. For instance, the \emph{invisible} search number (a.k.a. \emph{node} search number) of a graph is equal to its pathwidth plus~1, and the \emph{visible} search number of a graph is equal to its treewidth plus~1. The \emph{connected} variants of these games ask for search strategies that are connected, i.e., at every step of the strategy, the searched part of the graph induces a connected subgraph. We focus on \emph{monotone} search strategies, i.e., strategies for which every node is searched exactly once. The monotone connected visible search number of an $n$-node graph is at most $O(\log n)$ times its visible search number. First, we prove that this logarithmic bound is tight. Precisely, we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is $\Omega(\log n)$. Second, we prove that, as opposed to the non-connected variant of visible graph searching, ``recontamination helps" for connected visible search. Precisely, we prove that, for any $k \geq 4$, there exists a graph with connected visible search number at most $k$, and monotone connected visible search number $>k$.
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https://hal.inria.fr/inria-00423449
Contributor : Nicolas Nisse <>
Submitted on : Saturday, October 10, 2009 - 1:35:59 PM
Last modification on : Monday, October 12, 2020 - 10:30:12 AM

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  • HAL Id : inria-00423449, version 1

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Pierre Fraigniaud, Nicolas Nisse. Monotony Properties of Connected Visible Graph Searching. International Workshop on Graph-Theoretic Concepts in Computer Science (WG), 2006, Bergen, Norway. ⟨inria-00423449⟩

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