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Progress on Certifying Algorithms
"... A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying program P ( = the implementation of a certifying algorithm) inputs x, receives an output y and a ..."
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A certifying algorithm is an algorithm that produces with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying program P ( = the implementation of a certifying algorithm) inputs x, receives an output y and a certificate w, and then checks, either manually or by use of a checking program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust P. We refer the reader to the recent survey paper [9] for a detailed discussion of certifying algorithms. 1 An Example We illustrate the concept by an example. A matching in a graph G is a subset M of the edges of G such that no two share an endpoint. A matching has maximum cardinality if its cardinality is at least as large as that of any other matching. Figure 1 shows a graph and a maximum cardinality matching. Observe that the matching leaves two nodes unmatched, which gives rise to the question whether there exists a matching of larger cardinality. What is a witness for a matching being of maximum cardinality? Edmonds in his seminal papers [1,2] on how to compute maximum matchings in polynomial time introduced the following certificate: An oddset cover OSC of G is
CPlanarity of CConnected Clustered Graphs  Part I – Characterization
, 2006
"... We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2 ..."
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We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2] we exploit such a characterization to give a linear time cplanarity testing and embedding algorithm.
Monotone Drawings of Graphs
, 2012
"... We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected ..."
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We study a new standard for visualizing graphs: A monotone drawing is a straightline drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected planar graphs, we study the interplay between monotonicity, planarity, and convexity, and we outline a number of open problems and future research directions. Submitted:
CPlanarity of cconnected clustered graphs
, 2008
"... We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we ..."
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We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a lineartime cplanarity testing and embedding algorithm for cconnected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of wellknown data structures as SPQRtrees and BCtrees. If the test fails, the algorithm identifies a structural element responsible for the noncplanarity of the input clustered graph.
Contractions, Removals and How to Certify 3Connectivity in Linear Time
"... One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the exis ..."
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One of the most noted construction methods of 3vertexconnected graphs is due to Tutte and based on the following fact: Any 3vertexconnected graph G = (V, E) on more than 4 vertices contains a contractible edge, i. e., an edge whose contraction generates a 3connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K4, such that every intermediate graph is 3vertexconnected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(V  2) to O(E). This result has a number of consequences; an important one is a new lineartime test of 3connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in the last years. The test is conceptually different from wellknown lineartime 3connectivity tests and uses a certificate that is easy to verify in time O(E). We show how to extend the results to an optimal certifying test of 3edgeconnectivity. 1
Recognizing Outer 1Planar Graphs in Linear Time ⋆,⋆⋆
"... Abstract. A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard ..."
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Abstract. A graph is outer 1planar (o1p) if it can be drawn in the plane such that all vertices are on the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time and specialize 1planar graphs, whose recognition is N Phard. Our main result is a lineartime algorithm that first tests whether a graph G is o1p, and then computes an embedding. Moreover, the algorithm can augment G to a maximal o1p graph. If G is not o1p, then it includes one of six minors (see Fig. 3), which are also detected by the recognition algorithm. Hence, the algorithm returns a positive or negative witness for o1p. 1
Monotone Drawings of Graphs with Fixed Embedding ⋆
"... Abstract. A drawing of a graph is a monotone drawing if for every pair of vertices u and v, there is a path drawn from u to v that is monotone in some direction. In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as pa ..."
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Abstract. A drawing of a graph is a monotone drawing if for every pair of vertices u and v, there is a path drawn from u to v that is monotone in some direction. In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm. In this setting we prove that every planar graph on n vertices admits a planar monotone drawing with at most two bends per edge and with at most 4n − 10 bends in total; such a drawing can be computed in linear time and in polynomial area. We also show that two bends per edge are sometimes necessary on a linear number of edges of the graph. Furthermore, we investigate subclasses of planar graphs that can be realized as embeddingpreserving monotone drawings with straightline edges, and we show that biconnected embedded planar graphs and outerplane graphs always admit such drawings, which can be computed in linear time. 1
Finding 3Shredders Efficiently
, 2002
"... A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3shredder is a shredder of size three. We present an algorithm that, given a 3connected graph, finds its 3shredders in time proportional to the number of vertices and edges, when implement ..."
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A shredder in an undirected graph is a set of vertices whose removal results in at least three components. A 3shredder is a shredder of size three. We present an algorithm that, given a 3connected graph, finds its 3shredders in time proportional to the number of vertices and edges, when implemented on a RAM (random access machine).
Bend Minimization in Planar Orthogonal Drawings Using Integer Programming
, 2004
"... We consider the problem of minimizing the number of bends in a planar orthogonal graph drawing. While the problem can be solved via network flow for a given planar embedding of a graph G, it is NPhard if we consider the set of all planar embeddings of G. Our approach for biconnected graphs combines ..."
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We consider the problem of minimizing the number of bends in a planar orthogonal graph drawing. While the problem can be solved via network flow for a given planar embedding of a graph G, it is NPhard if we consider the set of all planar embeddings of G. Our approach for biconnected graphs combines an integer linear programming (ILP) formulation for the set of all embeddings of a planar graph with the network flow formulation for fixed embeddings. We report on extensive computational experiments with two benchmark sets containing a total of more than 12,000 graphs where we compared the performance of our ILPbased algorithm with a heuristic and a previously published branch & bound algorithm for solving the same problem. Our new algorithm is significantly faster than the previously published approach for the larger graphs of the benchmark graphs derived from industrial applications and almost twice as fast for the benchmark graphs from the artificially generated set of hard instances of the problem.