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### Page 1556 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews
The maximum genus of a relative graph (G, W) is the largest genus of an orientable surface upon which G admits a relative embedding with respect to W.  ...  The maximum genus of a graph is the largest genus of an ori- entable surface upon which G admits a cellular embedding.  ...

### Approximation algorithms for weighted matching

Shankar M. Venkatesan
1987 Theoretical Computer Science
Approximation algorithms are given for the maximum weighted matching problem in planar and other n-vertex graphs of genus g < n. For a planar graph, our approximation runs in O(min{n(  ...  Acknowledgment This paper is an extended and revised version of the results presented in [18] . The author thanks the referees for their helpful comments.  ...  Next, we show that an approximate maximum weighted matching in an n-vertex graph of genus g (g < n) can be found in 0( n3'2 log n +g(l/&)n log n) time given a drawing of the graph on a surface of genus  ...

### A Nebeský-Type Characterization for Relative Maximum Genus

Dan Archdeacon, C.Paul Bonnington, Jozef Širáň
1998 Journal of combinatorial theory. Series B (Print)
This paper concerns the maximum genus orientable surface upon which a given graph cellularly embeds. Classical theorems of Xuong and Nebesk y give exact values for the maximum genus.  ...  In this paper we establish the relative graph analogue of Nebesk y's Theorem.  ...  The minimum genus of a surface admitting a given graph is a di cult parameter. Here we focus primarily on the maximum genus of G, denoted M (G).  ...

### Page 2874 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews
We find a sufficient condition under which the relative genus and relative 2- cellular genus of the graph coincide.  ...  Summary (translated from the Russian): “Using the technique of graph rotations and certain well-known facts from the theory of the maximum genus of graphs, we show that for relative 2- cellular imbeddings  ...

### Simple greedy 2-approximation algorithm for the maximum genus of a graph [article]

Michal Kotrbcik, Martin Skoviera
2015 arXiv   pre-print
The maximum genus γ_M(G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding.  ...  This allows us to describe a greedy algorithm for the maximum genus of a graph; our algorithm returns an integer k such that γ_M(G)/2< k <γ_M(G), providing a simple method to efficiently approximate maximum  ...  Acknowledgement The first author was partially supported by Ministry of Education, Youth, and Sport of Czech Republic, Project No. CZ.1.07/2.3.00/30.0009.  ...

### Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph

Michal Kotrbcík, Martin Skoviera, Michael Wagner
2018 ACM-SIAM Symposium on Discrete Algorithms
The maximum genus γ M (G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding.  ...  As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong's combinatorial characterisation of maximum genus.  ...  The maximum genus γ M (G) parameter of a graph G is then the maximum integer g such that G has a cellular embedding in the orientable surface of genus g.  ...

### Upper-embeddable graphs and related topics

Nguyen Huy Xuong
1979 Journal of combinatorial theory. Series B (Print)
In this paper, we present different results concerning the structure of upperembeddable graphs. Various characterizations of graphs whose maximum genus satisfies additivity properties will be given.  ...  ACKNOWLEDGMENTS The author is grateful to A. Bouchet and C. Payan for helpful discussions.  ...  ON ADDITIVITY IN MAXIMUM GENUS IV.1 -Battle, Harary, Kodama and Youngs [l] have pointed out that the genus of a connected graph is the sum of the genera of its blocks.  ...

### Page 2792 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews
We then give an algorithm for obtaining the maximum genus of a graph on the basis of this theorem and find the maximum genus of a complete graph K, and a complete bigraph (which are already known) by this  ...  From the summary: “In this paper we first prove a theorem on finding the maximum genus of graphs.  ...

### Page 3160 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews
The author presents a formula for the maximum orientable genus of the surface on which a graph has a relative embedding with respect to a set W of closed walks.  ...  Paul (NZ-AUCK; Auckland) The relative maximum genus of a graph. (English summary) J. Combin. Theory Ser. B 60 (1994), no. 2, 195-206.  ...

### Cops and Robbers on Intersection Graphs [article]

Tomáš Gavenčiak, Przemysław Gordinowicz, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2016 arXiv   pre-print
The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G.  ...  and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g+15 in case of orientable surface of genus g, and at most 10g'+15 in case of non-orientable  ...  We note that it is proved in [4] that the maximum cop number of unit disk graphs is at most 9. Their strategy is similar to our strategy for string graphs, by applying Lemma 4.3.  ...

### Page 605 of Mathematical Reviews Vol. 50, Issue 3 [page]

1975 Mathematical Reviews
Zaks, Joseph 4363 The maximum genus of Cartesian products of graphs. Canad. J. Math, 26 (1974), 1025-1035. The maximum genus y,,(G) of a connected graph G is the maximum  ...  Ostroverhii, Hyper- relative genus of a graph (pp. 168-179); M. P. Homenko, M. O. Ostroverhii and V. O. Kuz’menko, Maximum genus of graphs (pp. 180-210); M. O. Ostroverhii and V. O.  ...

### Page 1445 of Mathematical Reviews Vol. , Issue 97C [page]

1997 Mathematical Reviews
new results give a complete picture of the tight lower bounds for the maximum genus of simplicial graphs.  ...  The authors show that the tight lower bound for the maximum genus of a 2-edge-connected simplicial graph is {8(G)/3), where B(G) is the cycle rank of graph G.  ...

### On the number of maximum genus embeddings of almost all graphs

Saul Stahl
1992 European journal of combinatorics (Print)
A lower bound for the number of maximum genus orientable embeddings of almost all graphs is provided.  ...  This bound is sufficiently strong to demonstrate that for those complete graphs that triangulate some orientable surface, the number of maximum genus embeddings far exceeds the number of minimum genus  ...  The purpose of this note is to substantiate the feeling, common amongst topological graph theorists, that most graphs have relatively many maximum genus embeddings.  ...

### An obstruction to embedding graphs in surfaces

Bojan Mohar
1989 Discrete Mathematics
It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.  ...  The main theorem also provides a convenient way for obtaining lower bounds on the genus, or the relative genus, of graphs.  ...  Ringel [9] proved that the maximum non-orientable genus of any connected graph G is equal to its Betti number B(G) = IE(G)I -IV(G)1 + 1.  ...

### SLICE: Enabling Greedy Routing in High Genus 3-D WSNs With General Topologies

Chen Wang, Hongbo Jiang, Tianlong Yu, John C. S. Lui
2016 IEEE/ACM Transactions on Networking
For a high genus 3-D volume network, SLICE embeds the inner nodes to a height structure attached to the convex polygon, and a variation of greedy routing scheme with guaranteed delivery is proposed in  ...  , resulting in a lower routing stretch. 2) This is the first work, to the best of our knowledge, that enables greedy routing in high genus 3-D WSNs with general topologies.  ...  Extracting a Maximum Cut Set For a genussurface , the first step is to find a maximum cut set , aiming at slicing to a genus-0 surface based on the Morse theory and Reeb graph.  ...
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