Faster Shortest-Path Algorithms for Planar Graphs

We propose data structures for maintaining shortest path in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3). Our data structure can be also applied to the problem of maintaining the maximum flow problem in an s − t planar network.

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter $\epsilon$ such that $0<\epsilon$ , our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a 1+ $\epsilon$ factor. The time bounds for both query and update for our algorithm is O( \epsilon -1 n 2/3 log 2 n log D) , where n is the number of nodes in G and D is the sum of its edge lengths. The time bound for the queries is worst case, while that for the additions is amortized. Our approximation algorithm is based upon a novel technique for approximately representing all-pairs shortest paths among a selected subset of the nodes by a sparse substitute graph.

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter e such that 0<e≤1, our algorithm maintains approximate allpairs shortest-paths in an undirected planar graph G with nonnegative edge lengths. The approximate paths are guaranteed to be accurate to within a (1+e)-factor. The time bounds for both query and update for our algorithm wis O(e−1n2/3 log2n log D), where n is the number of nodes in G and D is the sum of its edge lengths.

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = ( V , E ), one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed.Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service.Our results can be easily generalized to other wide classes of graphs---for instance we can take any minor-closed family of graphs.

Shortest Paths in Planar Graphs with Negative Weight Edges

Greenʼs theorem and isolation in planar graphs

We prove Nash‐Williams' conjecture that every 4‐connected, 3‐indivisible, infinite, planar graph contains a spanning 2‐way infinite path. A graph is said to be 3‐indivisible if the deletion of any finite set of vertices results in at most two infinite components. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 275–312, 2008

Improved induced matchings in sparse graphs

A generalization of P. Seymour's theorem on planar integral 2-commodity flows is given when the underlying graphG together with the demand graphH (a graph having edges that connect the corresponding terminal pairs) form a planar graph and the demand edges are on two faces ofG.

Let denote the maximum number of copies of in an vertex planar graph. The problem of bounding this function for various graphs has been extensively studied since the 70's. A special case that received a lot of attention recently is when is the path on vertices, denoted . Our main result in this paper is that This improves upon the previously best known bound by a factor , which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

We explore the restrictiveness of planarity on the complexity of computing the determinant and the permanent, and show that both problems remain as hard as in the general case, i.e. GapL and #P complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph.

The multiple pairs shortest path problem (MPSP) arises in many applications where the shortest paths and distances between only some specific pairs of origin-destination (OD) nodes in a network are desired. The traditional repeated single-source shortest path (SSSP) and all pairs shortest paths (APSP) algorithms often do unnecessary computation to solve the MPSP problem. We propose a new shortest path algorithm to save computational work when solving the MPSP problem. Our method is especially suitable for applications with fixed network topology but changeable arc lengths and desired OD pairs. Preliminary computational experiments demonstrate our algorithm’s superiority on airline network problems over other APSP and SSSP algorithms.

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

Motivated by an application in controllability we consider maximum matchings in random bipartite graphs G = (A, B). First we analyse Karp–Sipser's algorithm to determine the asymptotic size of maximum matchings in random bipartite graphs with a fixed degree distribution. We then allow an adversary to delete one edge adjacent to every vertex in A in the more restricted model where each vertex in A chooses d neighbours uniformly at random from B.