A Pseudo-approximation for the Genus of Hamiltonian Graphs

The genus of a graph is a very basic parameter in topological graph theory, that has been the subject of extensive study. Perhaps surprisingly, despite its importance, the problem of approximating the genus of a graph is very poorly understood. It has been shown to be NPcomplete by Thomassen [Tho89], and the best known upper bound for general graphs is an O(n)-approximation that follows by Euler’s characteristic.

[1]  Bruce A. Reed,et al.  A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[2]  Yury Makarychev,et al.  On graph crossing number and edge planarization , 2011, SODA '11.

[3]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[4]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[5]  Bojan Mohar,et al.  Face Covers and the Genus Problem for Apex Graphs , 2001, J. Comb. Theory, Ser. B.

[6]  Julia Chuzhoy,et al.  An algorithm for the graph crossing number problem , 2010, STOC '11.

[7]  Gary L. Miller,et al.  On determining the genus of a graph in O(v O(g)) steps(Preliminary Report) , 1979, STOC.

[8]  Markus Chimani,et al.  A Tighter Insertion-Based Approximation of the Crossing Number , 2011, ICALP.

[9]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

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

[11]  Bojan Mohar,et al.  A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface , 1999, SIAM J. Discret. Math..

[12]  Bojan Mohar,et al.  Generating locally cyclic triangulations of surfaces , 1992, J. Comb. Theory, Ser. B.

[13]  Markus Chimani,et al.  Approximating the crossing number of graphs embeddable in any orientable surface , 2010, SODA '10.

[14]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[15]  R. Ho.  Algebraic Topology , 2022 .

[16]  Carsten Thomassen.  Triangulating a Surface with a Prescribed Graph , 1993, J. Comb. Theory, Ser. B.

[17]  Markus Chimani,et al.  Approximating the Crossing Number of Apex Graphs , 2008, Graph Drawing.

[18]  Paul D. Seymour,et al.  Graph minors. VIII. A kuratowski theorem for general surfaces , 1990, J. Comb. Theory, Ser. B.

[19]  Jianer Chen,et al.  A Note on Approximating Graph Genus , 1997, Inf. Process. Lett..

[20]  Gareth Jones,et al.  GRAPHS OF GROUPS ON SURFACES: INTERACTIONS AND MODELS (North‐Holland Mathematics Studies 188) , 2003 .