The mixed search game against an agile and visible fugitive is monotone

Guillaume Mescoff, Christophe Paul, Dimitrios M. Thilikos

Research output: Contribution to journalArticlepeer-review


We consider the mixed search game against an agile and visible fugitive. This is the variant of the classic fugitive search game on graphs where searchers may be placed to (or removed from) the vertices or slide along edges. Moreover, the fugitive resides on the edges of the graph and can move at any time along unguarded paths. The mixed search number against an agile and visible fugitive of a graph G, denoted avms(G), is the minimum number of searchers required to capture a fugitive in this graph searching variant. Our main result is that this graph searching variant is monotone in the sense that the number of searchers required for a successful search strategy does not increase if we restrict the search strategies to those that do not permit the fugitive to visit an already “clean” edge. This means that mixed search strategies against an agile and visible fugitive can be polynomially certified, and therefore that the problem of deciding, given a graph G and an integer k, whether avms(G)≤k is in NP. Our proof is based on the introduction of the notion of tight bramble, that serves as an obstruction for the corresponding search parameter. Our results imply that for a graph G, avms(G) is equal to the Cartesian tree product number of G, that is, the minimum k for which G is a minor of the Cartesian product of a tree and a clique on k vertices.

Original languageEnglish (US)
Article number113345
JournalDiscrete Mathematics
Issue number4
StatePublished - Apr 2023


  • Bramble
  • Cartesian tree product number
  • Graph searching game
  • Tree decomposition

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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