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### A Linear-Time Algorithm for Finding Induced Planar Subgraphs

Shixun Huang, Zhifeng Bao, J. Shane Culpepper, Ping Zhang, Bang Zhang, Marc Herbstritt
2018 Symposium on Experimental and Efficient Algorithms
In contrast, we propose a linear-time algorithm that finds an induced planar subgraph of n − ν vertices in a graph of n vertices, where ν denotes the total number of vertices shared by the detected Kuratowski  ...  Edwards and Farr proposed an algorithm with O(mn) time complexity to find an induced planar subgraph of at least 3n/(d + 1) vertices in a graph of maximum degree d.  ...  x 6234567917 Linear-Time Algorithm for Finding Induced Planar SubgraphsAlgorithm 2: WalkdownWithSkips( Ḡ, v c ).  ...

### Approximation Algorithms for the Maximum Induced Planar and Outerplanar Subgraph Problems

Kerri Morgan, Graham Farr
2007 Journal of Graph Algorithms and Applications
The task of finding the largest subset of vertices of a graph that induces a planar subgraph is known as the Maximum Induced Planar Subgraph problem (MIPS).  ...  Efficient algorithms for finding large induced outerplanar graphs are also given.  ...  Acknowledgments We thank the referees for their helpful comments. K. Morgan et al., Algorithms for MIPS , JGAA, 11 (1) 165-193 (2007)  ...

### An Algorithm for Finding Large Induced Planar Subgraphs [chapter]

Keith Edwards, Graham Farr
2002 Lecture Notes in Computer Science
This paper presents an efficient algorithm that finds an induced planar subgraph of at least 3n/(d + 1) vertices in a graph of n vertices and maximum degree d.  ...  Our algorithm builds up an induced planar subgraph by iteratively adding a new vertex to it, or swapping a vertex in it with one outside it, in such a way that the procedure is guaranteed to stop, and  ...  In this paper, we present a polynomial time algorithm that finds an induced planar subgraph of at least 3n/(d + 1) vertices in a graph of maximum degree d.  ...

### Planarizing Graphs - A Survey and Annotated Bibliography

Annegret Liebers
2001 Journal of Graph Algorithms and Applications
Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph.  ...  While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature.  ...  [DT89] obtains an O(m log n) time algorithm for finding a maximal planar subgraph as a byproduct of an algorithm for incremental planarity testing.  ...

### Planar Induced Subgraphs of Sparse Graphs

Glencora Borradaile, David Eppstein, Pingan Zhu
2015 Journal of Graph Algorithms and Applications
These results are constructive, implying linear-time algorithms to find the respective induced subgraphs.  ...  We show that every graph has an induced pseudoforest of at least n-m/4.5 vertices, an induced partial 2-tree of at least n-m/5 vertices, and an induced planar subgraph of at least n-m/5.2174 vertices.  ...  The planarization problem Goal: find big planar subgraphs in nonplanar graphs Equivalently: delete as little as possible so the rest is planar In the version we study, the planar subgraphs are induced  ...

### Subgraph Isomorphism in Planar Graphs and Related Problems

David Eppstein
1999 Journal of Graph Algorithms and Applications
We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size.  ...  Plehn and Voigt  give an algorithm for subgraph isomorphism which in planar graphs takes time ), but this is still much larger than the linear bound we achieve.  ...  Acknowledgements The author wishes to thank Sandy Irani, George Lueker, and the anonymous referees, for helpful comments on drafts of this paper.  ...

### Subgraph Isomorphism in Planar Graphs and Related Problems [chapter]

David Eppstein
2002 Graph Algorithms and Applications I
We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size.  ...  Plehn and Voigt  give an algorithm for subgraph isomorphism which in planar graphs takes time ), but this is still much larger than the linear bound we achieve.  ...  Acknowledgements The author wishes to thank Sandy Irani, George Lueker, and the anonymous referees, for helpful comments on drafts of this paper.  ...

### Subgraph Isomorphism in Planar Graphs and Related Problems [article]

David Eppstein
1999 arXiv   pre-print
We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size.  ...  The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.  ...  Acknowledgements The author wishes to thank Sandy Irani, George Lueker, and the anonymous referees, for helpful comments on drafts of this paper.  ...

### Planar Induced Subgraphs of Sparse Graphs [chapter]

Glencora Borradaile, David Eppstein, Pingan Zhu
2014 Lecture Notes in Computer Science
These results are constructive, implying linear-time algorithms to find the respective induced subgraphs.  ...  We show that every graph has an induced pseudoforest of at least n − m/4.5 vertices, an induced partial 2-tree of at least n − m/5 vertices, and an induced planar subgraph of at least n − m/5.2174 vertices  ...  The authors thank Amir Nayyeri for helpful discussions.  ...

### Arboricity and bipartite subgraph listing algorithms

David Eppstein
1994 Information Processing Letters
In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs.  ...  The arboricity bound is necessary: for any constant k and any n there exists an n-vertex graph with O(n) edges and (n/ log n) k maximal complete bipartite subgraphs K k, .  ...  A decomposition of a planar graph into three forests can be found in linear time [7, 18] .  ...

### Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Jana Novotná, Karolina Okrasa, Michal Pilipczuk, Pawel Rzazewski, Erik Jan Van Leeuwen, Bartosz Walczak, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation
Consider the following problem: given a graph G ∈ D, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C.  ...  This leads, for example, to the following corollaries for specific classes C and D: a largest induced forest in a Pt-free graph can be found in 2Õ (n 2/3 ) time, for every fixed t; and a largest induced  ...  Algorithms for Max Induced Planar Subgraph and, more generally, Max Induced Σ-Embeddable Subgraph, were provided by Kociumaka and Pilipczuk  .  ...

### Parallel Planar Subgraph Isomorphism and Vertex Connectivity [article]

Lukas Gianinazzi, Torsten Hoefler
2020 arXiv   pre-print
By using a connection to certain separating cycles, our subgraph isomorphism algorithm can decide the vertex connectivity of a planar graph (with high probability) in asymptotically near-linear work and  ...  Our randomized low depth algorithm has a near-linear work dependency on the size of the target graph.  ...  Eppstein presents the first FPT subgraph isomorphism algorithm for planar graphs that has a linear dependency on the size of the pattern graph  .  ...

### An Alternative Method to Crossing Minimization on Hierarchical Graphs

Petra Mutzel
2001 SIAM Journal on Optimization
Based on a characterization of 2-level planar graphs, we give an integer linear programming formulation for the 2-level planarization problem.  ...  First, we give a motivation for our approach. Then, we address the problem of extracting a 2-level planar subgraph of maximum weight in a given 2-level graph. This problem is NP-hard.  ...  We build our hope on the fact that there is a linear time algorithm for recognizing h-level planar graphs (see [HP96] and [BN88] ).  ...

### On drawing a graph convexly in the plane (extended abstract) [chapter]

Hristo N. Djidjev
1995 Lecture Notes in Computer Science
We obtain a necessary and sufficient condition for the existence and a linear Mgorithm for the construction of such an extension.  ...  Let G be a planar graph and H be a subgraph of G. Given any convex drawing of H, we investigate the problem of how to extend the drawing of H to a convex drawing of G.  ...   construct a linear algorithm that finds an extendible outer faciM cycle of G by applying the algorithms for finding the triconnected components of G  and for planarity testing  on a graph  ...

### Page 7016 of Mathematical Reviews Vol. , Issue 89M [page]

1989 Mathematical Reviews
A linear decision tree queries COMPUTER SCIENCE 7016 “f(x) > 0” for a linear function f.  ...  Analogously, a graph G is said to be an induced minor of H if G is obtained from an induced subgraph of H by edge contraction.  ...
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