Mondshein Sequences (a.k.a. (2,1)-Orders)

Canonical orderings [STOC'88, FOCS'92] have been used as a key tool in graph drawing, graph encoding and visibility representations for the last decades. We study a far-reaching generalization of canonical orderings to non-planar graphs that was published by Lee Mondshein in a PhD-thesis at M.I.T. as early as 1971. Mondshein proposed to order the vertices of a graph in a sequence such that, for any i, the vertices from 1 to i induce essentially a 2-connected graph while the remaining vertices from i+1 to n induce a connected graph. Mondshein's sequence generalizes canonical orderings and became later and independently known under the name non-separating ear decomposition. Surprisingly, this fundamental link between canonical orderings and non-separating ear decomposition has not been established before. Currently, the fastest known algorithm for computing a Mondshein sequence achieves a running time of O(nm); the main open problem in Mondshein's and follow-up work is to improve this running time to subquadratic time. After putting Mondshein's work into context, we present an algorithm that computes a Mondshein sequence in optimal time and space O(m). This improves the previous best running time by a factor of n. We illustrate the impact of this result by deducing linear-time algorithms for five other problems, for four out of which the previous best running times have been quadratic. In particular, we show how to - compute three independent spanning trees of a 3-connected graph in time O(m), - improve the preprocessing time from O(n^2) to O(m) for a data structure reporting 3 internally disjoint paths between any given vertex pair, - derive a very simple O(n)-time planarity test once a Mondshein sequence has been computed, - compute a nested family of contractible subgraphs of 3-connected graphs in time O(m), - compute a 3-partition in time O(m).Comment: to appear in SIAM Journal on Computin

Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

Obtaining a Planar Graph by Vertex Deletion

In the k-Apex problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour, there is an O(n^3) time algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the big-O notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.Comment: 16 pages, 4 figures. A preliminary version of this paper appeared in the proceedings of WG 2007 (33rd International Workshop on Graph-Theoretic Concepts in Computer Science). The paper has been submitted to Algorithmic

Minimizing Movement: Fixed-Parameter Tractability

We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation, and mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general set up: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently.Comment: A preliminary version of the paper appeared in ESA 200

Inserting an Edge into a Geometric Embedding

The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs. Unfortunately, some graphs do not have a geometric embedding $\Gamma$ such that $\Gamma+e$ has the same number of crossings as the embedding $G+e$. This motivates the study of the computational complexity of the following problem: Given a combinatorially embedded graph $G$, compute a geometric embedding $\Gamma$ that has the same combinatorial embedding as $G$ and that minimizes the crossings of $\Gamma+e$. We give polynomial-time algorithms for special cases and prove that the general problem is fixed-parameter tractable in the number of crossings. Moreover, we show how to approximate the number of crossings by a factor $(\Delta-2)$, where $\Delta$ is the maximum vertex degree of $G$.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

The two-edge connectivity survivable-network design problem in planar graphs

Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum-cost subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem arising, for example, in telecommunications applications. We study a more general mixed-connectivity formulation, also employed in telecommunications optimization. Given a number (or requirement) r(v) in {0, 1, 2} for each vertex v in the graph, find a minimum-cost subgraph in which there are min{r(u), r(v)} edge-disjoint u-to-v paths for every pair u, v of vertices. We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is max SNP-hard in general graphs and strongly NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n log n). Under the additional restriction that the requirements are only non-zero for vertices on the boundary of a single face of a planar graph, we give a polynomial-time algorithm to find the optimal solution.Comment: Updated from original conference version (ICALP '08). To appear: Transactions on Algorithm

Simultaneous Embedding of Planar Graphs

Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.Comment: survey, 35 pages, 12 figure

Simple Recognition of Halin Graphs and Their Generalizations

We describe and implement two local reduction rules that can be used to recognize Halin graphs in linear time, avoiding the complicated planarity testing step of previous linear time Halin graph recognition algorithms. The same two rules can be used as the basis for linear-time algorithms for other algorithmic problems on Halin graphs, including decomposing these graphs into a tree and a cycle, finding a Hamiltonian cycle, or constructing a planar embedding. These reduction rules can also be used to recognize a broader class of polyhedral graphs. These graphs, which we call the D3-reducible graphs, are the dual graphs of the polyhedra formed by gluing pyramids together on their triangular faces; their treewidth is bounded, and they necessarily have Lombardi drawings.Comment: 16 pages, 5 figure

Disconnectivity and Relative Positions in Simultaneous Embeddings

The problem Simultaneous Embedding with Fixed Edges (SEFE) asks for two planar graph $G^1 = (V^1, E^1)$ and $G^2 = (V^2, E^2)$ sharing a common subgraph $G = G^1 \cap G^2$ whether they admit planar drawings such that the common graph is drawn the same in both. Previous results on this problem require $G$, $G^1$ and $G^2$ to be connected. This paper is a first step towards solving instances where these graphs are disconnected. First, we show that an instance of the general SEFE-problem can be reduced in linear time to an equivalent instance where $V^1 = V^2$ and $G^1$ and $G^2$ are connected. This shows that it can be assumed without loss of generality that both input graphs are connected. Second, we consider instances where $G$ is disconnected. We show that SEFE can be solved in linear time if $G$ is a family of disjoint cycles by introducing the CC-tree, which represents all simultaneous embeddings. We extend these results (including the CC-tree) to the case where $G$ consists of arbitrary connected components, each with a fixed embedding. Note that previous results require $G$ to be connected and thus do not need to care about relative positions of connected components. By contrast, we assume the embedding of each connected component to be fixed and thus focus on these relative positions. As SEFE requires to deal with both, embeddings of connected components and their relative positions, this complements previous work.Comment: 34 pages, 8 figure

Beyond the Euler characteristic: Approximating the genus of general graphs

Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an $O(1)$-approximation is not ruled out, the currently best-known upper bound is a trivial $O(n/g)$-approximation that follows from bounds on the Euler characteristic. In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we present a polynomial-time algorithm which given a graph $G$ of Euler genus $g$ outputs an embedding of $G$ into a surface of Euler genus $g^{O(1)}$. Combined with the above $O(n/g)$-approximation, our result also implies a $O(n^{1-\alpha})$-approximation, for some universal constant $\alpha>0$. Our approximation algorithm also has implications for the design of algorithms on graphs of small genus. Several of these algorithms require that an embedding of the graph into a surface of small genus is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding of the input graph is unknown