Search

It is proven that the connected pathwidth of any graph G is at most 2 · pw(G)+1, where pw(G) is the pathwidth of G. The method is constructive, i.e., it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k+1. The running time of the algorithm is O(dk²), where d is the number of "bags" in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality cs(G) ≤ 2s(G) + 3, where cs(G) and s(G) are the connected search number and the search number of G, respectively. Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a (monotone) connected one using 2k + 3 searchers and starting at an arbitrary homebase. [ABSTRACT FROM AUTHOR]

We define helicopter cops and robber games with multiple robbers, extending previous research, which considered only the pursuit of a single robber. Our model is defined for robbers that are visible (their position in the graph is known to the cops) and active (they can move at any point in the game) but is easily adapted to other variants of the single-robber game that have been considered in the literature. We show that the game with many robbers is nonmonotone: that is, fewer cops are needed if the robbers are allowed to reoccupy positions that were previously unavailable to them. As the moves of the cops depend on the position of the visible robbers, strategies for such games should be interactive, but the game becomes, in a sense, less interactive as the initial number of robbers increases. We prove that the main parameter emerging from the game, which we denote mvams(G, r), captures a hierarchy of parameters between proper pathwidth and proper treewidth, and we completely characterize it for trees, extending analogous existing characterizations of the pathwidth of trees. Moreover, we prove an upper bound for mvams(G, r) on general graphs and show that this bound is reached by an infinite class of graphs. On the other hand, if we consider the robbers to be invisible and lazy, the resulting parameters collapse in all cases to either proper pathwidth or proper treewidth, giving a further case where the classical equivalence between visible, active robbers and invisible, lazy robbers does not hold. [ABSTRACT FROM AUTHOR]