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An additivity theorem for the genus of a graph
1987
Journal of combinatorial theory. Series B (Print)
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The main goal of this paper is to prove a new additivity theorem for the genus of a graph. The theorem is true only by making a natural modification to the definition of the genus of an embedding. ...
Thus, computing the genus of a graph reduces to computing the genus of its Zconnected subgraphs. The additivity theorem [2] can be restated in terms of amalgams. DEFINITION. ...
doi:10.1016/0095-8956(87)90028-1
fatcat:azhiy647pbb5xnbofjahskunea
Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs
[chapter]
2009
Lecture Notes in Computer Science
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The first is an additive approximation for coloring within 2 of the optimal chromatic number, which is essentially best possible, and generalizes the seminal result by Thomassen [32] for bounded-genus ...
We show that this structure theorem is a powerful tool for developing algorithms on apex-minor-free graphs, including for the classic problems of coloring and TSP. ...
We thank Paul Seymour for helpful suggestions and intuition about the structure of apex-minor-free graphs. ...
doi:10.1007/978-3-642-02927-1_27
fatcat:en2uxhqyxraznbrc4cyucem64e
Upper-embeddable graphs and related topics
1979
Journal of combinatorial theory. Series B (Print)
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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. ...
In general, such an additivity property is not true for maximum genus. In the sequel, necessary and sufficient conditions for maximum genus additivity will be presented. ...
doi:10.1016/0095-8956(79)90059-5
fatcat:im4sqnsmz5hmtanqir6s4vp46m
Simple greedy 2-approximation algorithm for the maximum genus of a graph
[article]
2015
arXiv
pre-print
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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 ...
The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions. ...
arXiv:1501.07460v1
fatcat:tfabfjp5s5ghnakwtm46w4hqte
Minimal quadrangulations of surfaces
[article]
2021
arXiv
pre-print
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In this paper we determine n(Σ), the order of a minimal quadrangulation of a surface Σ, for all surfaces, both orientable and nonorientable. ...
A quadrangulation of Σ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in Σ. ...
The Euler characteristic of a surface Σ is denoted χ(Σ), which is 2 − 2g for S g , and 2 − q for N q . The Euler genus of Σ is defined as γ(Σ) = 2 − χ(Σ). ...
arXiv:2106.13377v1
fatcat:4cfflpilpngyppytf7x6m7kwdy
Improved Induced Matchings in Sparse Graphs
[chapter]
2009
Lecture Notes in Computer Science
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Theorem Every n-vertex twinless graph G of genus g contains a matching of size n+10(1−g )−1 7 . ...
Theorem Every n-vertex twinless graph G of genus g contains a matching of size n+10(1−g )−1 7 . ...
doi:10.1007/978-3-642-11269-0_11
fatcat:ytja66rr3ndq3dvbhkvuomi7ey
Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph
2018
ACM-SIAM Symposium on Discrete Algorithms
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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. ...
In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least γ M (G ...
Additionaly, we use ∪ for set union and G + e for the addition of an edge e to a graph G. ...
doi:10.4230/oasics.sosa.2019.14
dblp:conf/soda/KotrbcikS19
fatcat:ox222ht6grdttipdz7crzqrg3a
Page 3959 of Mathematical Reviews Vol. , Issue 88h
[page]
1988
Mathematical Reviews
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Summary: “The main goal of this paper is to prove a new additivity theorem for the genus of a graph. ...
—8y)'/*], where x is the Euler characteristic of S.”
88h:05036 05C10
Miller, Gary L. (1-SCA-C)
An additivity theorem for the genus of a graph. ...
Topological directions in Cops and Robbers
[article]
2018
arXiv
pre-print
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We survey results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface ...
After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus g, then its cop number is at most g + 3. ...
The authors gratefully acknowledge support from NSERC. The second author is also supported in part by the Canada Research Chairs program, and by the Research Grant P1-0297 of ARRS (Slovenia). ...
arXiv:1709.09050v2
fatcat:nug57vbzc5ezxf75bvryhfni7m
Face distributions of embeddings of complete graphs
[article]
2018
arXiv
pre-print
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We exhibit such an embedding for each complete graph except K_8, the complete graph on 8 vertices, and we go on to prove that no such embedding can exist for this graph. ...
Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e. the numbers of faces of each length) realizable by minimum genus embeddings ...
The construction for Theorem 13.7 takes an embedding of K 2t+2 with two extra edges and glues it, without any additional augmentation, to a triangular embedding of another graph, so with the same construction ...
arXiv:1708.02092v3
fatcat:4jfhq7rab5c3bbpn55uwdukioa
Page 3390 of Mathematical Reviews Vol. , Issue 88g
[page]
1988
Mathematical Reviews
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For a graph G, the Euler genus e(G) of G is the smallest Euler genus among all surfaces in which G embeds. The following additivity theorem is proved. ...
Summary: “In earlier works, additivity theorems for the genus and Euler genus of unions of graphs at two points have been given. ...
An additivity theorem for maximum genus of a graph
1978
Discrete Mathematics
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If G is the union of two blocks, then a necessary and sufficiernt condition is given for the maximum genus of G to be the sum of the maximum genera of its blocks. ...
If in addition the blocks of G are upper embeddable, then a necessary and sufficient condition is given for the upper embeddability of G. ...
Let N bc a graph with components G1, G2,. . . , G, and G a connected gruph obtainec: from H! by ihe addition of n -1 edges. Then In this paper we refer to the Betti nuw:ber, F(G), of a graph G. ...
doi:10.1016/0012-365x(78)90148-6
fatcat:knttfqfxinblzac3ahfoz5ocg4
Linearity of grid minors in treewidth with applications through bidimensionality
2008
Combinatorica
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We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. ...
This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. ...
Seymour for many helpful discussions and for providing a portal into the Graph Minor Theory and revealing some of its hidden structure that we use in this paper. ...
doi:10.1007/s00493-008-2140-4
fatcat:cff2tl55u5bubhmz65rewhxpoa
A Kuratowski-type theorem for the maximum genus of a graph
1972
Journal of combinatorial theory. Series B (Print)
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A graph G, is said to be a subdivision of G if G, can be obtained from G in a finite number of steps, each consisting of the deletion of an edge uv and the addition of a new vertex w together with the ...
The parameters y(G) and yM(G) are cahe the genus and the maximum genus of the graph 6, respectively. ...
doi:10.1016/0095-8956(72)90040-8
fatcat:zwofsouw3berlhxbbfefwofn7e
Low complexity algorithms in knot theory
[article]
2018
arXiv
pre-print
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Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We show that the genus problem for these knots belongs to TC^0 circuit complexity class. ...
We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace(n). ...
By the Theorem 3.5 the genus of an alternating knot K is equal to the genus of an alternating diagram of K. ...
arXiv:1803.04908v2
fatcat:tli55rg6i5aydoubuw5up4uupi
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