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SCHNYDER DECOMPOSITIONS FOR REGULAR PLANE GRAPHS AND APPLICATION TO DRAWING
"... Abstract. Schnyder woods are decompositions of simple triangulations into three edgedisjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to dangulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposi ..."
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Abstract. Schnyder woods are decompositions of simple triangulations into three edgedisjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to dangulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d − 2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the dangulation is d. As in the case of Schnyder woods (d = 3), there are alternative formulationsintermsoforientations (“fractional ” orientations when d ≥ 5)and in terms of cornerlabellings. Moreover, the set of Schnyder decompositions on a fixed dangulation of girth d is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on dregular plane graphs of mincut d rooted at a vertex v ∗ ) are decompositions into d spanning trees rooted atv ∗ such that each edge not incidentto v ∗ isused in opposite directions by two trees. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d = 4, these correspond to wellstudied structures on simple quadrangulations (2orientations and partitions into 2 spanning trees). In the case d = 4, the dual of even Schnyder decompositions yields (planar) orthogonal and straightline drawing algorithms. For a 4regular plane graph G of mincut 4 with n vertices plus a marked vertex v, the vertices of G\v are placed on a (n−1)×(n−1) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−2 edges of G\v has exactly one bend. Embedding also the marked vertex v is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to v. We propose a further compaction step for the drawing algorithm and show that the obtained gridsize is strongly concentrated around 25n/32 ×25n/32 for a uniformly random instance with n vertices. 1.
BIJECTIVE COUNTING OF INVOLUTIVE BAXTER PERMUTATIONS
"... Abstract. We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size 2n with no fixed points is 3·2 n−1 (n+1)(n+2) ..."
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Abstract. We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size 2n with no fixed points is 3·2 n−1 (n+1)(n+2)
On the Characterization of Plane Bus Graphs
"... Abstract. Bus graphs are being used for the visualization of hyperedges, for example in VLSI design. Formally, they are specified by bipartite graphs G = (B ∪ V, E) of bus vertices B realized by single horizontal and vertical segments, and point vertices V that are connected orthogonally to the bus ..."
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Abstract. Bus graphs are being used for the visualization of hyperedges, for example in VLSI design. Formally, they are specified by bipartite graphs G = (B ∪ V, E) of bus vertices B realized by single horizontal and vertical segments, and point vertices V that are connected orthogonally to the bus segments. The decision whether a bipartite graph admits a bus realization is NPcomplete. In this paper we show that in contrast the question whether a plane bipartite graph admits a planar bus realization can be answered in polynomial time. We first identify three necessary conditions on the partition B = BV ∪ BH of the bus vertices, here BV denotes the vertical and BH the horizontal buses. We provide a test whether good partition, i.e., a partition obeying these conditions, exist. The test is based on the computation of maximum matching on some auxiliary graph. Given a good partition we can construct a noncrossing realization of the bus graph on an O(n) × O(n) grid in linear time. 1
Exploiting AirPressure to Map Floorplans on Point Sets
"... Abstract. We prove a conjecture of Ackerman, Barequet and Pinter. Every floorplan with n segments can be embedded on every set of n points in generic position. The construction makes use of area universal floorplans also known as area universal rectangular layouts. The notion of area used in our con ..."
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Abstract. We prove a conjecture of Ackerman, Barequet and Pinter. Every floorplan with n segments can be embedded on every set of n points in generic position. The construction makes use of area universal floorplans also known as area universal rectangular layouts. The notion of area used in our context depends on a nonuniform density function. We, therefore, have to generalize the theory of area universal floorplans to this situation. The method is then used to prove a result about accommodating points in floorplans that is slightly more general than the conjecture of Ackerman et al.
Involutions on Baxter Objects
"... Abstract. Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a for ..."
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Abstract. Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge’s “q = −1 phenomenon”. Résumé. Les nombres Baxter comptent plusieurs familles d’objets combinatoires, qui sont tous équipées avec des involutions naturels. Dans ce papier, nous ajoutons une famille combinatoire à la liste, et nous montrons que les bijections connus entre ces objets respectent ces involutions. En plus, nous donnons une formule pour le nombre d’objets fixés par cette involution et nous montrons qu’elle est une instance du “phénomène q = −1 ” de Stembridge.
More Combinatorics of Fulton’s Essential Set Masato Kobayashi
"... Copyright c ○ 2013 Masato Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We develop combinatorics of Fulton’s essential s ..."
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Copyright c ○ 2013 Masato Kobayashi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We develop combinatorics of Fulton’s essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret ErikssonLinusson’s characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of BruhatEhresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.