Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call ℎ-Clique2Path Planarity: Given a graph G, whose vertices are partitioned into subsets of size at most h, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of G is planar. We study this problem when G is a simple topological graph, and establish its complexity in relation to k-planarity. We prove that ℎ-Clique2Path Planarity is NP-complete even when ℎ=4 and G is a simple 3-plane graph, while it can be solved in linear time, for any h, when G is 1-plane.
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Quasipolynomiality of the Smallest Missing Induced Subgraph
We study the problem of finding the smallest graph that does not occur as an induced subgraph of a given graph. This missing induced subgraph has at most logarithmic size and can be found by a brute-force search, in an $n$-vertex graph, in time $n^{O(\log n)}$. We show that under the Exponential Time Hypothesis this quasipolynomial time bound is optimal. We also consider variations of the problem in which either the missing subgraph or the given graph comes from a restricted graph family; for instance, we prove that the smallest missing planar induced subgraph of a given planar graph can be found in polynomial time.
more » « less- Award ID(s):
- 2212129
- PAR ID:
- 10545600
- Publisher / Repository:
- Brown
- Date Published:
- Journal Name:
- Journal of Graph Algorithms and Applications
- Volume:
- 27
- Issue:
- 5
- ISSN:
- 1526-1719
- Page Range / eLocation ID:
- 329-339
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.7155/jgaa.00625
