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Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 32 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
Drawings of planar graphs with few slopes and segments
 Computational Geometry Theory and Applications 38:194–212
, 2005
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2 ..."
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Cited by 15 (4 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2n segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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Cited by 4 (3 self)
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we
Really straight drawings I: Planar graphs
, 2005
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/ ..."
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Cited by 1 (1 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Geometric Shellings of 3Polytopes
, 1999
"... A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3polytopes. Recently, a graph theoretical characterizatio ..."
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Cited by 1 (0 self)
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A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3polytopes. Recently, a graph theoretical characterization of geometric shellings of 3polytopes were given by Holt & Klee and Mihalisin & Klee. ffl We first give a characterization of shellings of 3polytopes. Then we show sufficient conditions for a shellings to be geometric: ffl the first and the last facet being adjacent, ffl any facet (except the first two) being adjacent to no less than two previous facets or ffl the induced orders being geometric shellings for two smaller polytopes made by dividing the polytope at a triple of facets adjacent to each other but not sharing a vertex. Simple 3polytopes allow perturbations of facets, thus may have more chance a shelling is geometric. As sufficient conditions for this case we show: ffl the ...
Recoloring Directed Graphs
"... Let G be a directed graph and k a positive integer. We define the kcolor graph of G (Dk(G) for short) as the directed graph having all kcolorings of G as node set, and where two kcolorings β and ϕ are joined by a directed edge β → ϕ if ϕ is obtained from β by choosing a vertex v and recoloring v ..."
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Let G be a directed graph and k a positive integer. We define the kcolor graph of G (Dk(G) for short) as the directed graph having all kcolorings of G as node set, and where two kcolorings β and ϕ are joined by a directed edge β → ϕ if ϕ is obtained from β by choosing a vertex v and recoloring v so that its color is different from the colors of all its outneighbors. We investigate reachability questions in Dk(G). In particular we want to know whether a fixed legal k ′coloring ψ of G with k ′ ≤ k is reachable in Dk(G) from every possible initial kcoloring β. Interesting instances of this problem arise when G is planar and the orientation is an arbitrary αorientation for fixed α. Our main result is that reachability can be guaranteed if the orientation has maximal outdegree ≤ k − 1 and an accessible pseudosink. 1
Straight Line Triangle Representations
"... Abstract. Which plane graphs admit a straight line representations such that all faces have the shape of a triangle? We present a characterization based on flat angle assignements, i.e., selections of angles of the graph that have size π in the representation. Another characterization is in terms of ..."
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Abstract. Which plane graphs admit a straight line representations such that all faces have the shape of a triangle? We present a characterization based on flat angle assignements, i.e., selections of angles of the graph that have size π in the representation. Another characterization is in terms of contact systems of pseudosegments. We use discrete harmonic functions to show that contact systems of pseudosegments that respect certain conditions are stretchable. The drawback of the characterization is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable. 1
Geometric Shellings of 3Polytopes 3
, 1999
"... A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3polytopes. Recently, a graph theoretical characterizatio ..."
Abstract
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A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3polytopes. Recently, a graph theoretical characterization of geometric shellings of 3polytopes were given by Holt & Klee and Mihalisin & Klee. We Then we show su the rst give acharacterization of shellings of 3polytopes. cient conditions for a shellings to be geometric: rst and the last facet being adjacent, any facet (except the rst two) being adjacent to no less than two previous facets or the induced orders being geometric shellings for two smaller polytopes made by dividing the polytope at a triple of facets adjacent to each other but not sharing a vertex. Simple 3polytopes allow perturbations of facets, thus may havemore chance a shelling is geometric. As su cient conditions for this case we show: the induced order being a geometric shelling for a smaller polytope made by removing a triangular or a quadrilateral facet or joining two consecutive facets in a shelling or the polytope only having triangular or quadrilateral facets. A nongeometric shelling of a (simplicial) 3polytope was rst shown by Smilansky. We show such example for a simple 3polytope, which is minimal with respect to the number of facets. The discussions proceed in the polar setting: as total orders of vertices of the polar polytope. All of our main results can be stated in graph theoretical terms. 1