Results 1  10
of
17
The LCMlattice in monomial resolutions
, 1999
"... Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: ..."
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Cited by 38 (5 self)
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Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were:
Planar Graphs As Minimal Resolutions Of Trivariate Monomial Ideals
, 2001
"... We introduce a new kind of plane drawing for simple triconnected planar graphs, called rigid embeddings in grid surfaces. They provide methods to (1) solve the problem of nding wellstructured (cellular, in this case) minimal free resolutions for arbitrary monomial ideals in three variables; (2) ..."
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Cited by 15 (1 self)
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We introduce a new kind of plane drawing for simple triconnected planar graphs, called rigid embeddings in grid surfaces. They provide methods to (1) solve the problem of nding wellstructured (cellular, in this case) minimal free resolutions for arbitrary monomial ideals in three variables; (2) give a new proof of the BrightwellTrotter bound on the order dimension of triconnected planar maps; and (3) generalize Schnyder's angle coloring of planar triangulations to arbitrary triconnected planar maps. Rigid embedding is stable under duality for planar maps, and has certain uniqueness properties.
The order dimension of planar maps
 SIAM J. DISCRETE MATH
, 1997
"... This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion, ..."
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Cited by 12 (4 self)
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This is a sequel to a previous paper entitled The Order Dimension of Convex Polytopes, by the same authors [SIAM J. Discrete Math., 6 (1993), pp. 230–245]. In that paper, we considered the poset PM formed by taking the vertices, edges, and faces of a 3connected planar map M, ordered by inclusion, and showed that the order dimension of PM is always equal to 4. In this paper, we show that if M is any planar map, then the order dimension of PM is still at most 4.
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Graphs and partially ordered sets: recent results and new directions
 JACOBSON (EDS.), SURVEYS IN GRAPH THEORY, CONGRESSUS NUMERANTIUM
, 1996
"... We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each ..."
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Cited by 9 (2 self)
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We survey some recent research progress on topics linking graphs and finite partially ordered sets. Among these topics are planar graphs, hamiltonian cycles and paths, graph and hypergraph coloring, online algorithms, intersection graphs, inclusion orders, random methods and ramsey theory. In each case, we discuss open problems and future research directions.
Geodesic Embeddings and Planar Graphs
, 2002
"... Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better u ..."
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Cited by 9 (6 self)
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Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove a characterization of 3connected planar graphs as those graphs admitting rigid geodesic embeddings, a bijection between Schnyder labelings and rigid geodesic embeddings, a strong version of the BrightwellTrotter theorem.
Posets and planar graphs
 JOURNAL OF GRAPH THEORY
, 2000
"... Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [t − 1 ↕t] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [2↕3]. ..."
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Cited by 7 (6 self)
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Usually dimension should be an integer valued parameter, we introduce a refined version of dimension for graphs which can assume a value [t − 1 ↕t] which is thought to be between t − 1 and t. We have the following two results: • A graph is outerplanar if and only if its dimension is at most [2↕3]. This characterization of outerplanar graphs is closely related to the celebrated result of W. Schnyder [16] who proved that a graph is planar if and only if its dimension is at most 3. • The largest n for which the dimension of the complete graph Kn is at most [t − 1↕t] is the number of antichains in the lattice of all subsets of a set of size t − 2. Accordingly, the refined dimension problem for complete graphs is equivalent to the classical combinatorial problem known as Dedekind’s problem. This result extends work of Ho¸sten and Morris [14]. The main results are enriched by background material which links to a line of reserch in extremal graph theory which was stimulated by a problem posed by G. Agnarsson: Find the maximum number of edges in a graph on n nodes with dimension at most t.
Schnyder woods and orthogonal surfaces
 In Proceedings of Graph Drawing
, 2006
"... In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated ..."
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Cited by 7 (3 self)
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In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the BrightwellTrotter Theorem which says, that the face lattice of a 3polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time. 1
Bijections for Baxter Families and Related Objects
, 2008
"... The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2) ..."
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Cited by 7 (5 self)
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The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)
Adjacency posets of planar graphs
 DISCRETE MATH
"... In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show t ..."
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Cited by 4 (3 self)
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In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dimension of the adjacency poset of a planar bipartite graph is at most 4. This result is best possible. More generally, the dimension of the adjacency poset of a graph is bounded as a function of its genus and so is the dimension of the vertexface poset of such a graph.