A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs

Given a graph G=(V,E) and two vertices s,t ∈ V , s\neq t , the Menger problem is to find a maximum number of disjoint paths connecting s and t . Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or arc disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the arc disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time \bf O (|V| log |V|) . Here we present a linear time, i.e., asymptotically optimal, algorithm for the arc disjoint version in planar directed graphs. Received August 1997; revised January 1999.     

Optimal Reconstruction of Graphs under the Additive Model

We study the problem of reconstructing unknown graphs under the additive combinatorial search model. The main result concerns the reconstruction of bounded degree graphs, i.e., graphs with the degree of all vertices bounded by a constant d . We show that such graphs can be reconstructed in O(dn) nonadaptive queries, which matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of d -separating matrices, which settles an open question stated by Lindstr?m in [20]. Finally, we consider several particular classes of graphs. We show how an optimal nonadaptive solution of O(n ² / log n) queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a nonadaptive algorithm. Received August 1997; revised January 1999.     

Drawing Planar Graphs Symmetrically, II: Biconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals the structure in the graph. This paper discusses symmetric drawings of biconnected planar graphs. More specifically, we discuss geometric automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a drawing of G. Finding geometric automorphisms is the first and most difficult step in constructing symmetric drawings of graphs. The problem of determining whether a given graph has a non-trivial geometric automorphism is NP-complete for general graphs. In this paper we present a linear time algorithm for finding planar geometric automorphisms of biconnected planar graphs. A drawing algorithm is also discussed.     

Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Dynamically Switching Vertices in Planar Graphs

We consider graphs whose vertices may be in one of two different states: either on or off . We wish to maintain dynamically such graphs under an intermixed sequence of updates and queries. An update may reverse the status of a vertex, by switching it either on or off , and may insert a new edge or delete an existing edge. A query tests whether any two given vertices are connected in the subgraph induced by the vertices that are on . We give efficient algorithms that maintain information about connectivity on planar graphs in O( log ³ n) amortized time per query, insert, delete, switch-on, and switch-off operation over sequences of at least Ω(n) operations, where n is the number of vertices of the graph. Received September 1997; revised January 1999.     

Partitioning Planar Graphs with Vertex Costs: Algorithms and Applications

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total sum of vertex costs at most and that this bound is optimal to within a constant factor. Moreover, such a separator can be found in linear time. This theorem implies a variety of other separation results. We describe applications of our separator theorems to graph embedding problems, to graph pebbling, and to multicommodity flow problems. Received June 1997; revised February 1999.     

An Algorithm for Straight-Line Drawing of Planar Graphs

We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs ``nicely,' as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n-4) x (n-2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Received September 21, 1995; revised March 15, 1996.     

Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

The maximum linear arrangement problem for trees under projectivity and planarity

A linear arrangement is a mapping π from the n vertices of a graph G to n distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.     

Parallel Algorithms for Series Parallel Graphs and Graphs with Treewidth Two 1

In this paper a parallel algorithm is given that, given a graph G=(V,E) , decides whether G is a series parallel graph, and, if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log \kern -1pt |E|log ^\ast \kern -1pt |E|) time and O(|E|) operations on an EREW PRAM, and O(log \kern -1pt |E|) time and O(|E|) operations on a CRCW PRAM. The results hold for undirected as well as for directed graphs. Algorithms with the same resource bounds are described for the recognition of graphs of treewidth two, and for constructing tree decompositions of treewidth two. Hence efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs and graphs with treewidth two. These include many well-known problems like all problems that can be stated in monadic second-order logic. Received July 15, 1997; revised January 29, 1999, and June 23, 1999.     

Disk Embeddings of Planar Graphs

Given a planar graph $G=(V,E)$ and a rooted forest ${\FF}=(V_{\FF}, A_{\FF})$ with leaf set $V$, we wish to decide whether $G$ has a plane embedding $\GG$ satisfying the following condition: There are $|V_{\FF}|-|V|$ pairwise noncrossing Jordan curves in the plane one-to-one corresponding to the nonleaf vertices of ${\FF}$ such that for every nonleaf vertex $f$ of ${\FF}$, the interior of the curve $\JJ_f$ corresponding to $f$ contains all the leaf descendants of $f$ in ${\FF}$ but contains no other leaves of ${\FF}$. This problem arises from theoretical studies in geographic database systems. It is unknown whether this problem can be solved in polynomial time. This paper presents an almost linear-time algorithm for a nontrivial special case where the set of leaf descendants of each nonleaf vertex $f$ in ${\FF}$ induces a connected subgraph of $G$.     

Separators in Graphs with Negative and Multiple Vertex Weights

A separator theorem for a class of graphs asserts that every graph in the class can be divided approximately in half by removing a set of vertices of specified size. Nontrivial separator theorems hold for several classes of graphs, including graphs of bounded genus and chordal graphs. We show that any separator theorem implies various weighted separator theorems. In particular, we show that if the vertices of the graph have real-valued weights, which may be positive or negative, then the graph can be divided exactly in half according to weight. If k unrelated sets of weights are given, the graph can be divided simultaneously by all k sets of weights. These results considerably strengthen earlier results of Gilbert, Lipton, and Tarjan: (1) for k=1 with the weights restricted to being nonnegative, and (2) for k>1 , nonnegative weights, and simultaneous division within a factor of (1+ε ) of exactly in half. Received November 21, 1995; revised October 30, 1997.     

Planar packing of trees and spider trees

In [A. García, C. Hernando, F. Hurtado, M. Noy, J. Tejel, Packing trees into planar graphs, J. Graph Theory (2002) 172-181] García et al. conjectured that for every two non-star trees there exists a planar graph containing them as edge-disjoint subgraphs. In this paper we prove the conjecture in the case in which one of the trees is a spider tree.     

A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks. Given an error parameter such that , 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+ factor. The time bounds for both query and update for our algorithm is O( ^-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. Received January 1995; revised February 1997.     

一种简单的图的平面性判断算法的实现研究

Malgrange、Malgrange和Pertuiset三人合作提出O（n2）时间复杂度的平面性判断算法,尽管效率不是那么理想,却易于理解,并且算法结束时能够给出平面图的一种平面嵌入,另外算法仅涉及到割点的检测、图的计算机表示、图的分割、图的遍历等较为基础的问题。从而能够很好地适应教学及入门对直观性,可实现性的需要。尽管这个方法已经较为直观。但是由于图的平面嵌入在计算机中的表示较为困难等问题．其算法具体如何实现依然需要细心研究。     

On Embedding a Graph in the Grid with the Maximum Number of Bends and Other Bad Features

Graph Drawing is (usually) concerned with the production of readable representations of graphs. In this paper, instead of investigating how to produce “good” drawings we tackle the opposite problem of producing “bad” drawings. In particular, we study how to construct orthogonal drawings with many bends along the edges and with large area. Our results show surprising contact points, in Graph Drawing, between the computational cost of niceness and the one of ugliness.     

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

We present a fixed-parameter algorithm that constructively solves the $k$-dominating set problem on any class of graphs excluding a single-crossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-minor-free graphs and the $K_{5}$-minor-free graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set, and a collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.     

Triangulating input-constrained planar point sets

We present a linear-time algorithm for computing a triangulation of n points in 2D whose positions are constrained to n disjoint disks of uniform size, after O(nlogn) preprocessing applied to these disks. Our algorithm can be extended to any collection of convex sets of bounded areas and aspect ratios, assuming no point lies in more than some constant number of sets (bounded depth of overlap), and each set contains only a constant number of query points.     

On Coloring Unit Disk Graphs

In this paper the coloring problem for unit disk (UD) graphs is considered. UD graphs are the intersection graphs of equal-sized disks in the plane. Colorings of UD graphs arise in the study of channel assignment problems in broadcast networks. Improving on a result of Clark et al. [2] it is shown that the coloring problem for UD graphs remains NP-complete for any fixed number of colors k≥ 3 . Furthermore, a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques. Received February 12, 1996; revised October 9, 1996.     

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

We present an algorithm that takes I/Os (sort(N)=Θ((N/(DB))log  _M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N, where k is a constant. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in  I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/O-efficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take I/Os. The maximal matching algorithm is used in the tree decomposition algorithm. An abstract of this paper was presented at the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pp. 89–90, 2001. Research of A. Maheshwari supported by NSERC. Part of this work was done while the second author was a Ph.D. student at the School of Computer Science of Carleton University.