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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

1987
*
Theoretical Computer Science
*

Approximation algorithms are given for

doi:10.1016/0304-3975(87)90022-3
fatcat:7z6dwfqo35g77j4xguv6e4wsz4
*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

1998
*
Journal of combinatorial theory. Series B (Print)
*

This paper concerns

doi:10.1006/jctb.1997.1810
fatcat:avgoavp4svg6jniiqc5pojf4na
*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]

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

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

1979
*
Journal of combinatorial theory. Series B (Print)
*

In this paper, we present different results concerning

doi:10.1016/0095-8956(79)90059-5
fatcat:im4sqnsmz5hmtanqir6s4vp46m
*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]

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

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

1989
*
Discrete Mathematics
*

It is shown that

doi:10.1016/0012-365x(89)90170-2
fatcat:7wpseo2xszbw3l7e42vixcz6qq
*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

2016
*
IEEE/ACM Transactions on Networking
*

For

doi:10.1109/tnet.2015.2464312
fatcat:iwefmc4mtzezhke47qlwnxwbde
*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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