Adler, I orcid.org/0000-0002-9667-9841, Paul, C and Thilikos, DM (2021) Connected Search for a Lazy Robber. Journal of Graph Theory, 97 (4). pp. 510-552. ISSN 0364-9024
Abstract
The node-search game against a lazy (or, respectively, agile) invisible robber has been introduced as a search-game analogue of the treewidth parameter (and, respectively, pathwidth). In the connected variants of the above games, we additionally demand that, at each moment of the search, the clean territories are connected. The connected search game against an agile and invisible robber has been extensively examined. The monotone variant (where we demand that the clean territories are progressively increasing) of this game corresponds to the graph parameter of connected pathwidth. It is known that the price of connectivity to search for an agile robber is bounded by 2, that is, the connected pathwidth of a graph is at most twice (plus some constant) its pathwidth. We investigate the study of the connected search game against a lazy robber. A lazy robber moves only when the cops' strategy threatens the vertex that he currently occupies. We introduce two alternative graph-theoretical formulations of this game, one in terms of connected tree-decompositions and one in terms of (connected) layouts, leading to the graph parameter of connected treewidth. We observe that the connected treewidth parameter is closed under contractions and prove that for every k ≥ 2 , the set of contraction obstructions of the class of graphs with connected treewidth at most k is infinite. Our main result is a complete characterization of the obstruction set for k = 2 . We also show that, in contrast to the agile robber game, the price of connectivity is unbounded.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 Wiley Periodicals LLC. This is the peer reviewed version of the following article: Adler, I., Paul, C., and Thilikos, D. M., Connected search for a lazy robber, J. Graph Theory. 2021; 97: 510– 552, which has been published in final form at https://doi.org/10.1002/jgt.22669. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. |
Keywords: | connected treewidth; contraction obstructions; cops and robbers game; graph searching; price of connectivity |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 12 Mar 2021 17:09 |
Last Modified: | 21 Mar 2022 01:38 |
Status: | Published |
Publisher: | Wiley |
Identification Number: | 10.1002/jgt.22669 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:172074 |