Results 1  10
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11
Genus distributions of cubic outerplanar graphs
, 2011
"... We present a quadratictime algorithm for computing the genus distribution of any 3regular outerplanar graph. Although recursions and some formulas for genus distributions have previously been calculated for bouquets and for various kinds of ladders and other special families of graphs, cubic outer ..."
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Cited by 12 (12 self)
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We present a quadratictime algorithm for computing the genus distribution of any 3regular outerplanar graph. Although recursions and some formulas for genus distributions have previously been calculated for bouquets and for various kinds of ladders and other special families of graphs, cubic outerplanar graphs now emerge as the most general family of graphs whose genus distributions are known to be computable in polynomial time. The key algorithmic features are the syntheses of the given outerplanar graph by a sequence of edgeamalgamations of some of its subgraphs, in the order corresponding to the postorder traversal of a plane tree that we call the inner tree, and the coordination of that synthesis with justintime rootsplitting.
Genus distribution of graph amalgamations: Pasting at rootvertices
 ARS COMBINATORIA 94
, 2010
"... We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We int ..."
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Cited by 11 (11 self)
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We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of faceboundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broadbased approach to calculating genus distributions for graphs synthesized from smaller graphs.
Genus distributions of graphs under selfedgeamalgamations
 Ars Math. Contemporanea
"... We investigate the wellknown problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two rootedges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these rootedges be know ..."
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Cited by 8 (8 self)
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We investigate the wellknown problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two rootedges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these rootedges be known and that both rootedges have two 2valent endpoints. We derive general formulas for calculating the genus distributions of graphs that can be obtained either by selfcoamalgamating or by selfcontraamalgamating a base graph whose partitioned genus distribution is already known. We see how these general formulas provide a unified approach to calculating genus distributions of many new graph families, such as copasted and contrapasted closed chains of copies of the triangular prism graph, as well as graph families like circular and Möbius ladders with previously known solutions to the genus distribution problem.
Genus distribution of 4regular outerplanar graphs
, 2011
"... We present an O(n 2)time algorithm for calculating the genus distribution of any 4regular outerplanar graph. We characterize such graphs in terms of what we call split graphs and incidence trees. The algorithm uses postorder traversal of the incidence tree and productions that are adapted from a ..."
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Cited by 7 (7 self)
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We present an O(n 2)time algorithm for calculating the genus distribution of any 4regular outerplanar graph. We characterize such graphs in terms of what we call split graphs and incidence trees. The algorithm uses postorder traversal of the incidence tree and productions that are adapted from a previous paper that analyzes doubleroot vertexamalgamations and selfamalgamations.
Embeddings of Graphs of Fixed Treewidth and Bounded Degree
, 2013
"... Let F be any family of graphs of fixed treewidth and bounded degree. We construct a quadratictime algorithm for calculating the genus distribution of the graphs in F. Within a postorder traversal of the decomposition tree, the algorithm involves a fullpowered upgrading of production rules and roo ..."
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Cited by 4 (4 self)
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Let F be any family of graphs of fixed treewidth and bounded degree. We construct a quadratictime algorithm for calculating the genus distribution of the graphs in F. Within a postorder traversal of the decomposition tree, the algorithm involves a fullpowered upgrading of production rules and rootpopping. This algorithm for calculating genus distributions in quadratic time complements an algorithm of Kawarabayashi, Mohar, and Reed for calculating the minimum genus of a graph of bounded treewidth in linear time.
Genus Distribution of P3✷Pn
"... We derive a recursion for the genus distribution of the graph family P3✷Pn, with the aid of a modified collection of doubleroot partials. We introduce a new kind of production, which corresponds to a surgical operation more complicated than the vertex or edgeamalgamation operations used in our ea ..."
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We derive a recursion for the genus distribution of the graph family P3✷Pn, with the aid of a modified collection of doubleroot partials. We introduce a new kind of production, which corresponds to a surgical operation more complicated than the vertex or edgeamalgamation operations used in our earlier work. 1
GENUS DISTRIBUTIONS OF STARLADDERS
"... Abstract. Starladder graphs were introduced by Gross in his development of a quadratictime algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of starladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials. Newly ..."
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Abstract. Starladder graphs were introduced by Gross in his development of a quadratictime algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of starladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials. Newly developed methods have led to a number of recent papers that derive genus distributions and total embedding distributions for various families of graphs. Our focus here is on a family of graphs called starladders. 1.
Genus Distributions of Iterated 3Wheels and 3Prisms
 ARS MATHEMATICA CONTEMPORANEA
"... The iterated 3prism P r n 3 is the cartesian product C3✷Pn of a 3cycle and an nvertex path. At each end of the iterated 3prism, there is a 3cycle whose vertices are 3valent in C3✷Pn. The iterated 3wheel W n 3 is obtained by contracting one of these 3cycles in C3✷Pn+1 to a single vertex. Usin ..."
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The iterated 3prism P r n 3 is the cartesian product C3✷Pn of a 3cycle and an nvertex path. At each end of the iterated 3prism, there is a 3cycle whose vertices are 3valent in C3✷Pn. The iterated 3wheel W n 3 is obtained by contracting one of these 3cycles in C3✷Pn+1 to a single vertex. Using rootedgraphs, we derive simultaneous recursions for and a formula for the genus distribution of the the partitioned genus distributions of W n 3 graphs P rn 3. A seemingly straightforward way to construct either the sequence of iterated prisms P rn 3 or the sequence of iterated wheels W n 3, would be by iterative amalgamation of a copy of C3✷K2, such that a copy of C3 contained in it is matched to the “newest ” copy of C3 in the growing graph. Calculating genus distributions for the sequences would then involve an excessively large set of simultaneous recurrences. To avoid this, we propose a method of iterative surgery, under which the same vertex is considered a rootvertex in all graphs of the sequence, and in which the successive calculations of genus distributions require only four simultaneous recurrences. We also prove that the genus distribution of P rn 3 not only dominates the genus distribution of W n−1 3, but is also dominated by the genus distribution of W n 3.