We study the problem of computing semantic-preserving word clouds in which semantically related w... more We study the problem of computing semantic-preserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is associated with rectangle with fixed dimensions, and the goal is to represent semantically related words by ensuring that the two corresponding rectangles touch. We design and analyze efficient polynomial-time algorithms for some variants of the WRAC problem, show that several general variants are NP-hard, and describe a number of approximation algorithms. Finally, we experimentally demonstrate that our theoretically-sound algorithms outperform the early heuristics.
We consider the problem of covering a host graph G with several graphs from a fixed template clas... more We consider the problem of covering a host graph G with several graphs from a fixed template class T . The classical covering number of G with respect to T is the minimum number of template graphs needed to cover the edges of G. We introduce two new parameters: the local and the folded covering number. Each parameter measures how far G is from the template class in a different way. Whereas the folded covering number has been investigated thoroughly for some template classes, e.g., interval graphs and planar graphs, the local covering number was given only little attention. We provide new bounds on each covering number w.r.t. the following template classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters turning up this way are interval-number, track-number, and linear-, star-, and caterpillar arboricity. As host graphs we consider graphs of bounded degeneracy, bounded degree, or bounded (simple) tree-width, as well as, outerplanar, planar bipartite and planar graphs. For several pairs of a host class and a template class we determine the maximum (local, folded) covering number of a host graph w.r.t. that template class exactly.
We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, eac... more We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, each build their own path by claiming one vertex at a time, starting with Alice. The vertices carry non-negative weights that sum up to 1 and either player tries to claim a path with larger total weight than the opponent. We show that if the graph is a tree then Alice can always ensure to get at most 1/5 less than Bob, and that there exist trees where Bob can ensure to get at least 1/5 more than Alice.
We consider the non-crossing connectors problem, which is stated as follows: Given n simply conne... more We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R 1 , . . . , R n in the plane and finite point sets P i ⊂ R i for i = 1, . . . , n, are there non-crossing connectors γ i for (R i , P i ), i.e., arc-connected sets
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge int... more The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.
We consider L-graphs, that is contact graphs of axis-aligned L-shapes in the plane, all with the ... more We consider L-graphs, that is contact graphs of axis-aligned L-shapes in the plane, all with the same rotation. We provide several characterizations of Lgraphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L's can always be converted to an equivalent one with equilateral L's. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of square-based cuboids. We also study a slightly more restricted version of equilateral L-contact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs. We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem.
A mixed hypergraph is a triple H = (V, C, D), where V is a set of vertices, C and D are sets of h... more A mixed hypergraph is a triple H = (V, C, D), where V is a set of vertices, C and D are sets of hyperedges. A vertex-coloring of H is proper if C-edges are not totally multicolored and D-edges are not monochromatic. The feasible set S(H) of H is the set of all integers, s, such that H has a proper coloring with s colors.
We study on-line colorings of certain graphs given as intersection graphs of objects "between two... more We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(ω 3 ) colors on graphs with maximum clique size ω.
Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in... more Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120-133] asked whether there are two non-isomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Nešetřil et al. showed that any two graphs with different clique number [Combinatorica 1(2) (1981), 199-202] or different odd girth [Comment.
Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they char... more Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling.
ABSTRACT We study contact representations of graphs in which vertices are represented by axis-ali... more ABSTRACT We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra.
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come... more An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, L , L and L . A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an L, an L or L , a k-bend path, or a segment, then this graph is called an {L}-graph, {L, L }-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1): [365][366][367][368][369][370][371][372] 1992], stating that every {L, L }-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are {L}-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {L}-graphs. Furthermore we show that all complements of planar graphs are B19-VPG-graphs and all complements of full subdivisions are B2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.
We study the problem of computing semantic-preserving word clouds in which semantically related w... more We study the problem of computing semantic-preserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is associated with rectangle with fixed dimensions, and the goal is to represent semantically related words by ensuring that the two corresponding rectangles touch. We design and analyze efficient polynomial-time algorithms for some variants of the WRAC problem, show that several general variants are NP-hard, and describe a number of approximation algorithms. Finally, we experimentally demonstrate that our theoretically-sound algorithms outperform the early heuristics.
We consider the problem of covering a host graph G with several graphs from a fixed template clas... more We consider the problem of covering a host graph G with several graphs from a fixed template class T . The classical covering number of G with respect to T is the minimum number of template graphs needed to cover the edges of G. We introduce two new parameters: the local and the folded covering number. Each parameter measures how far G is from the template class in a different way. Whereas the folded covering number has been investigated thoroughly for some template classes, e.g., interval graphs and planar graphs, the local covering number was given only little attention. We provide new bounds on each covering number w.r.t. the following template classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters turning up this way are interval-number, track-number, and linear-, star-, and caterpillar arboricity. As host graphs we consider graphs of bounded degeneracy, bounded degree, or bounded (simple) tree-width, as well as, outerplanar, planar bipartite and planar graphs. For several pairs of a host class and a template class we determine the maximum (local, folded) covering number of a host graph w.r.t. that template class exactly.
We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, eac... more We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, each build their own path by claiming one vertex at a time, starting with Alice. The vertices carry non-negative weights that sum up to 1 and either player tries to claim a path with larger total weight than the opponent. We show that if the graph is a tree then Alice can always ensure to get at most 1/5 less than Bob, and that there exist trees where Bob can ensure to get at least 1/5 more than Alice.
We consider the non-crossing connectors problem, which is stated as follows: Given n simply conne... more We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R 1 , . . . , R n in the plane and finite point sets P i ⊂ R i for i = 1, . . . , n, are there non-crossing connectors γ i for (R i , P i ), i.e., arc-connected sets
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge int... more The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.
We consider L-graphs, that is contact graphs of axis-aligned L-shapes in the plane, all with the ... more We consider L-graphs, that is contact graphs of axis-aligned L-shapes in the plane, all with the same rotation. We provide several characterizations of Lgraphs, drawing connections to Schnyder realizers and canonical orders of maximally planar graphs. We show that every contact system of L's can always be converted to an equivalent one with equilateral L's. This can be used to show a stronger version of a result of Thomassen, namely, that every planar graph can be represented as a contact system of square-based cuboids. We also study a slightly more restricted version of equilateral L-contact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs. We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem.
A mixed hypergraph is a triple H = (V, C, D), where V is a set of vertices, C and D are sets of h... more A mixed hypergraph is a triple H = (V, C, D), where V is a set of vertices, C and D are sets of hyperedges. A vertex-coloring of H is proper if C-edges are not totally multicolored and D-edges are not monochromatic. The feasible set S(H) of H is the set of all integers, s, such that H has a proper coloring with s colors.
We study on-line colorings of certain graphs given as intersection graphs of objects "between two... more We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(ω 3 ) colors on graphs with maximum clique size ω.
Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in... more Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120-133] asked whether there are two non-isomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Nešetřil et al. showed that any two graphs with different clique number [Combinatorica 1(2) (1981), 199-202] or different odd girth [Comment.
Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they char... more Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling.
ABSTRACT We study contact representations of graphs in which vertices are represented by axis-ali... more ABSTRACT We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra.
An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come... more An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, L , L and L . A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an L, an L or L , a k-bend path, or a segment, then this graph is called an {L}-graph, {L, L }-graph, B k -VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer [Discrete Mathematics, 108(1): [365][366][367][368][369][370][371][372] 1992], stating that every {L, L }-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are {L}-graphs, or B k -VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {L}-graphs. Furthermore we show that all complements of planar graphs are B19-VPG-graphs and all complements of full subdivisions are B2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.
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