1,623 research outputs found
Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II
Let be a connected graph with and with the spanning
forest . Let be a real number and let be a real function. In this paper, we show that if for all
, , then has a spanning tree
containing such that for each vertex , , where
denotes the number of components of and denotes the
number of edges of with both ends in . This is an improvement of several
results and the condition is best possible. Next, we also investigate an
extension for this result and deduce that every -edge-connected graph
has a spanning subgraph containing edge-disjoint spanning trees such
that for each vertex , , where ; also if contains
edge-disjoint spanning trees, then can be found such that for each vertex
, , where .
Finally, we show that strongly -tough graphs, including -tough
graphs of order at least three, have spanning Eulerian subgraphs whose degrees
lie in the set . In addition, we show that every -tough graph has
spanning closed walk meeting each vertex at most times and prove a
long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed
walk; connected factor; toughness; total exces
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
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