Journal of Graph Algorithms and Applications Genus Distributions of Cubic Outerplanar Graphs

We present a quadratic-time algorithm for computing the genus distribution of any 3-regular 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 edge-amalgamations of some of its subgraphs, in the order corresponding to the post-order traversal of a plane tree that we call the inner tree, and the coordination of that synthesis with just-in-time root-splitting.

[1]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[2]  Mike J. Grannell,et al.  Exponential Families of Non-Isomorphic Triangulations of Complete Graphs , 2000, J. Comb. Theory, Ser. B.

[3]  Marston D. E. Conder,et al.  Determination of all Regular Maps of Small Genus , 2001, J. Comb. Theory, Ser. B.

[4]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[5]  T. Walsh,et al.  Counting rooted maps by genus II , 1972 .

[6]  Jonathan L. Gross,et al.  Genus distribution of graphs under surgery: adding edges and splitting vertices , 2010 .

[7]  Bojan Mohar,et al.  Embedding graphs in an arbitrary surface in linear time , 1996, STOC '96.

[8]  Jonathan L. Gross,et al.  Genus distributions of graphs under self-edge-amalgamations , 2012, Ars Math. Contemp..

[9]  Jonathan L. Gross,et al.  Embeddings of cubic Halin graphs: Genus distributions , 2012, Ars Math. Contemp..

[10]  Yanpei Liu,et al.  Orientable embedding distributions by genus for certain type of non-planar graphs (I) , 2006, Ars Comb..

[11]  Jonathan L. Gross,et al.  Genus distributions of graphs under edge-amalgamations , 2010, Ars Math. Contemp..

[12]  Richard Statman,et al.  Genus distributions for two classes of graphs , 1989, J. Comb. Theory, Ser. B.

[13]  Saul Stahl,et al.  Permutation-partition pairs. III. Embedding distributions of linear families of graphs , 1991, J. Comb. Theory, Ser. B.

[14]  Saul Stahl,et al.  Region distributions of some small diameter graphs , 1991, Discret. Math..

[15]  Saul Stahl,et al.  Region distributions of graph embeddings and stirling numbers , 1990, Discret. Math..

[16]  Liu Yan-pei,et al.  Genus Distribution for Two Classes of Graphs , 2006 .

[17]  Esther Hunt Tesar.  Genus distribution of Ringel ladders , 2000, Discret. Math..

[18]  Jonathan L. Gross,et al.  Genus distribution of graph amalgamations: Pasting when one root has arbitrary degree , 2010, Ars Math. Contemp..

[19]  Jin Ho Kwak,et al.  Total embedding distributions for bouquets of circles , 2002, Discret. Math..

[20]  Jonathan L. Gross,et al.  Hierarchy for imbedding-distribution invariants of a graph , 1987, J. Graph Theory.

[21]  Robert Cori,et al.  Counting Non-Isomorphic Chord Diagrams , 1998, Theor. Comput. Sci..

[22]  Tao Wang,et al.  The Total Embedding Distributions of Cacti and Necklaces , 2006 .

[23]  Jonathan L. Gross,et al.  Genus distributions for bouquets of circles , 1989, J. Comb. Theory, Ser. B.

[24]  Jonathan L. Gross,et al.  Genus Distribution of Graph Amalgamations: Pasting at Root-Vertices , 2010, Ars Comb..

[25]  D. Jackson.  Counting cycles in permutations by group characters, with an application to a topological problem , 1987 .

[26]  T. Walsh,et al.  Counting rooted maps by genus III: Nonseparable maps , 1975 .

[27]  Carsten Thomassen,et al.  The Graph Genus Problem is NP-Complete , 1989, J. Algorithms.

[28]  Jozef Sirán,et al.  Triangular embeddings of complete graphs from graceful labellings of paths , 2007, J. Comb. Theory B.

[29]  David M. Jackson,et al.  An atlas of the smaller maps in orientable and nonorientable surfaces , 2000 .

[30]  Vladimir P. Korzhik,et al.  Exponential Families of Non-isomorphic Non-triangular Orientable Genus Embeddings of Complete Graphs , 2002, J. Comb. Theory, Ser. B.

[31]  Terry I. Visentin,et al.  On the Genus Distribution of (p, q, n)-Dipoles , 2007, Electron. J. Comb..

[32]  D. Jackson,et al.  A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus , 1990 .

[33]  Jonathan L. Gross,et al.  Topological Graph Theory , 1987, Handbook of Graph Theory.

[34]  Yanpei Liu,et al.  Orientable embedding genus distribution for certain types of graphs , 2008, J. Comb. Theory, Ser. B.

[35]  J. Harer,et al.  The Euler characteristic of the moduli space of curves , 1986 .

[36]  Jonathan L. Gross,et al.  Genus distribution of graph amalgamations: self-pasting at root-vertices , 2011, Australas. J Comb..

[37]  Lyle Andrew Mcgeoch,et al.  Algorithms for two graph problems: computing maximum-genus imbeddings and the two-server problem , 1987 .

[38]  Jonathan L. Gross,et al.  Genus Distributions of 4-Regular Outerplanar Graphs , 2011, Electron. J. Comb..

[39]  John Beidler,et al.  Data Structures and Algorithms , 1996, Wiley Encyclopedia of Computer Science and Engineering.

[40]  S. Gottwald,et al.  Fuzzy set theory and its applications. Second edition , 1992 .

[41]  L. Beineke,et al.  Topics in Topological Graph Theory , 2009 .

[42]  Mike J. Grannell,et al.  A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs , 2008, J. Comb. Theory, Ser. B.

[43]  Jin Ho Kwak,et al.  Enumeration of graph embeddings , 1994, Discret. Math..