5 research outputs found
From Pathwidth to Connected Pathwidth
It is proven that the connected pathwidth of any graph is at most
2\cdot\pw(G)+1, where \pw(G) is the pathwidth of . The method is
constructive, i.e. it yields an efficient algorithm that for a given path
decomposition of width computes a connected path decomposition of width at
most . The running time of the algorithm is , where 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
\csn(G)\leq 2\sn(G)+3, where \csn(G) and \sn(G) are the connected search
number and the search number of . Moreover, the algorithm presented in this
work can be used to convert a given search strategy using searchers into a
(monotone) connected one using searchers and starting at an arbitrary
homebase
Connected searching of weighted trees
AbstractIn this paper we consider the problem of connected edge searching of weighted trees. Barrière et al. claim in [L. Barrière, P. Flocchini, P. Fraigniaud, N. Santoro, Capture of an intruder by mobile agents, in: SPAA’02: Proceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, NY, USA, 2002, pp. 200–209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in their algorithm, the problem turns out to be open. It is proven in this paper that the considered problem is strongly NP-complete even for node-weighted trees (the weight of each edge is 1) with one vertex of degree greater than 2. It is also shown that there exists a polynomial-time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices. This is an FPT algorithm with respect to the maximum degree of a tree