General Bounds on the Downhill Domination Number in Graphs.

A path π = (v1, v2,...vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 \u3c i \u3c k, deg(vi) \u3e deg(vi+1), where deg(vi) denotes the degree of vertex vi ∊ V. The downhill domination number equals the minimum cardinality of a set S ⊂ V having the property that every vertex v ∊ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is at most half its order, and that the downhill domination number of a tree is at most one third its order. We characterize the graphs obtaining each of these bounds

Uphill & Downhill Domination in Graphs and Related Graph Parameters.

Placing degree constraints on the vertices of a path allows the definitions of uphill and downhill paths. Specifically, we say that a path π = v1, v2,...vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(vi+1). Conversely, a path π = u1, u2,...uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(ui) ≤ deg(ui+1). We investigate graphical parameters related to downhill and uphill paths in graphs. For example, a downhill path set is a set P of vertex disjoint downhill paths such that every vertex v ∈ V belongs to at least one path in P, and the downhill path number is the minimum cardinality of a downhill path set of G. For another example, the downhill domination number of a graph G is defined to be the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We determine relationships among these invariants and other graphical parameters related to downhill and uphill paths. We also give a polynomial time algorithm to find a minimum downhill dominating set and a minimum uphill dominating set for any graph

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Degree monotone paths and graph operations

A path P in a graph G is said to be a degree monotone path if the sequence of degrees of the vertices of P in the order in which they appear on P is monotonic. The length of the longest degree monotone path in G is denoted by mp(G). This parameter was first studied in an earlier paper by the authors where bounds in terms of other parameters of G were obtained. In this paper we concentrate on the study of how mp(G) changes under various operations on G. We first consider how mp(G) changes when an edge is deleted, added, contracted or subdivided. We similarly consider the effects of adding or deleting a vertex. We sometimes restrict our attention to particular classes of graphs. Finally we study mp(G × H) in terms of mp(G) and mp(H) where × is either the Cartesian product or the join of two graphs. In all these cases we give bounds on the parameter mp of the modified graph in terms of the original graph or graphs and we show that all the bounds are sharppeer-reviewe