Contraction obstructions for connected graph searching

Micah J. Best, Arvind Gupta, Dimitrios M. Thilikos, Dimitris Zoros

Research output: Contribution to journalArticlepeer-review


We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for k=2, where k is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the “sense of direction” of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite.

Original languageEnglish (US)
Pages (from-to)27-47
Number of pages21
JournalDiscrete Applied Mathematics
StatePublished - Aug 20 2016


  • Graph contractions
  • Graph searching
  • Obstruction set

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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