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

Sweeping Graphs and Digraphs

Searching a network for an intruder is an interesting and difficult problem. Sweeping is one such search model, in which we "sweep" for intruders along edges. The minimum number of sweepers needed to clear a graph G is known as the sweep number sw(G). The sweep number can be restricted by insisting the sweep be monotonic (once an edge is cleared, it must stay cleared) and connected (new clear edges must be incident with already cleared edges). We will examine several lower bounds for sweep number, among them minimum degree, clique number, chromatic number, and girth. We will make use of several of these bounds to calculate sweep numbers for several infinite families of graphs. In particular, these families will answer some open problems regarding the relationships between the monotonic sweep number, connected sweep number, and monotonic connected sweep number. While sweeping originated in simple graphs, the idea may be easily extended to directed graphs, which allow for four different sweep models. We will examine some interesting non-intuitive sweep numbers and look at relations between these models. We also look at bounds on these sweep numbers on digraphs and tournaments

Boundary-optimal Triangulation Flooding

Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence of the triangles a flooding. Our main result shows that it is in general impossible to guarantee boundary size O(n1-ffl), for any ffl> 0, and a number of regions that is o(log n), where n is the number of faces of the triangulation. We also show that a flooding with boundary size O(pn) and O(log n) regions can be computed in O(n log n) time