LIFO-search: A min–max theorem and a searching game for cycle-rank and tree-depth

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Abstract

We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We show that all common variations of the game require the same number of searchers. We then introduce the notion of (directed) shelters in (di)graphs and prove a min–max theorem implying their equivalence to the cycle-rank/tree-depth parameter in (di)graphs. As (directed) shelters provide escape strategies for the fugitive, this implies that the LIFO-search game is monotone and that the LIFO-search parameter is equivalent to the one of cycle-rank/tree-depth in (di)graphs.

Keywords

Graph parameters
Graph searching
Pursuit–evasion games
Cycle-rank
Tree-depth
Obstructions
Min–max theorem

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Part of this research was presented during Dagstuhl Seminar 11071 (13.02.11–18.02.11) on Theory and Applications of Graph Searching Problems (Fomin et al., 2011) [11]. Preliminary versions of this paper appeared in EUROCOMB 2011 (Giannopoulou and Thilikos, 2011) [13] and in FCT 2011 (Hunter, 2011) [15].