1,068 research outputs found
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
Size of the Largest Induced Forest in Subcubic Graphs of Girth at least Four and Five
In this paper, we address the maximum number of vertices of induced forests
in subcubic graphs with girth at least four or five. We provide a unified
approach to prove that every 2-connected subcubic graph on vertices and
edges with girth at least four or five, respectively, has an induced forest on
at least or vertices, respectively, except
for finitely many exceptional graphs. Our results improve a result of Liu and
Zhao and are tight in the sense that the bounds are attained by infinitely many
2-connected graphs. Equivalently, we prove that such graphs admit feedback
vertex sets with size at most or , respectively.
Those exceptional graphs will be explicitly constructed, and our result can be
easily modified to drop the 2-connectivity requirement
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
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