7,844 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
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs
A greedy embedding of a graph into a metric space is a
function such that in the embedding for every pair of
non-adjacent vertices there exists another vertex adjacent
to which is closer to than . This notion of greedy
embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci.
2005), where authors conjectured that every 3-connected planar graph has a
greedy embedding (possibly planar and convex) in the Euclidean plane. Recently,
greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008).
However, their algorithm do not result in a drawing that is planar and convex
for all 3-connected planar graph in the Euclidean plane. In this work we
consider the planar convex greedy embedding conjecture and make some progress.
We derive a new characterization of planar convex greedy embedding that given a
3-connected planar graph , an embedding x: V \to \bbbr^2 of is
a planar convex greedy embedding if and only if, in the embedding , weight
of the maximum weight spanning tree () and weight of the minimum weight
spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq
(\card{V}-1)^{1 - \delta}, for some .Comment: 19 pages, A short version of this paper has been accepted for
presentation in FCT 2009 - 17th International Symposium on Fundamentals of
Computation Theor
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
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