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### An additivity theorem for the genus of a graph

Gary L Miller
1987 Journal of combinatorial theory. Series B (Print)
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.  ...

### Approximation Algorithms via Structural Results for Apex-Minor-Free Graphs [chapter]

Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
2009 Lecture Notes in Computer Science
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.  ...

### 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.  ...  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.  ...

### 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  ...  The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions.  ...

### Minimal quadrangulations of surfaces [article]

Wenzhong Liu, M. N. Ellingham, Dong Ye
2021 arXiv   pre-print
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).  ...

### Improved Induced Matchings in Sparse Graphs [chapter]

Rok Erman, Łukasz Kowalik, Matjaž Krnc, Tomasz Waleń
2009 Lecture Notes in Computer Science
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 .  ...

### 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.  ...  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.  ...

### Page 3959 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews
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]

Anthony Bonato, Bojan Mohar
2018 arXiv   pre-print
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).  ...

### Face distributions of embeddings of complete graphs [article]

Timothy Sun
2018 arXiv   pre-print
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  ...

### Page 3390 of Mathematical Reviews Vol. , Issue 88g [page]

1988 Mathematical Reviews
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

C.H.C. Little, R.D. Ringeisen
1978 Discrete Mathematics
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.  ...

### Linearity of grid minors in treewidth with applications through bidimensionality

2008 Combinatorica
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.  ...

### A Kuratowski-type theorem for the maximum genus of a graph

E.A Nordhaus, R.D Ringeisen, B.M Stewart, A.T White
1972 Journal of combinatorial theory. Series B (Print)
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.  ...

### Low complexity algorithms in knot theory [article]

Olga Kharlampovich, Alina Vdovina
2018 arXiv   pre-print
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.  ...
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