Algorithmic complexity and extremality characterizations for edge searching and its variations

Abstract

Edge searching is a combinatorial game played on graphs. The aim is to construct a search strategy to catch an intruder hidden in the graph independent of his actions. If the intruder has a diffused form then searching corresponds to cleaning the graph. A related problem consists of minimizing the number of searchers used in this search. Various versions of edge searching have been introduced in the past depending on how searchers and the intruder can move. In this dissertation we define Weighted Search and Fast Search as two new variants and answer some complexity and extremality problems. -- Weighted Search corresponds to cleaning a contaminated graph where edges may have different capacities. The main result we have is that Weighted Search is an NP-complete problem. We also give comparison results such as bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize those graphs which two searchers can clean. -- Fast Search is an internal monotone search where no edge is traversed more than once in a non-weighted graph. We present a linear time algorithm to compute a fast search strategy for a given tree. We investigate the fast search strategies for bipartite graphs. -- The construction of k-searchable graphs, those graphs which k searchers can clean, has been of major interest. Graphs that are 1,2 or 3-searchable have been completely characterized previously, whereas characterizing 4-searchable graphs was left as an open problem. We solve this problem partially and give insights for future work.