1,422 research outputs found
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Some extremal problems for hereditary properties of graphs
This note answers extremal questions like: what is the maximum number of
edges in a graph of order n, which belongs to some hereditary property. The
same question is answered also for the spectral radius and other similar
parameters
VoG: Summarizing and Understanding Large Graphs
How can we succinctly describe a million-node graph with a few simple
sentences? How can we measure the "importance" of a set of discovered subgraphs
in a large graph? These are exactly the problems we focus on. Our main ideas
are to construct a "vocabulary" of subgraph-types that often occur in real
graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the
most succinct description of a graph in terms of this vocabulary. We measure
success in a well-founded way by means of the Minimum Description Length (MDL)
principle: a subgraph is included in the summary if it decreases the total
description length of the graph.
Our contributions are three-fold: (a) formulation: we provide a principled
encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop
\method, an efficient method to minimize the description cost, and (c)
applicability: we report experimental results on multi-million-edge real
graphs, including Flickr and the Notre Dame web graph.Comment: SIAM International Conference on Data Mining (SDM) 201
Clique decompositions of multipartite graphs and completion of Latin squares
Our main result essentially reduces the problem of finding an
edge-decomposition of a balanced r-partite graph of large minimum degree into
r-cliques to the problem of finding a fractional r-clique decomposition or an
approximate one. Together with very recent results of Bowditch and Dukes as
well as Montgomery on fractional decompositions into triangles and cliques
respectively, this gives the best known bounds on the minimum degree which
ensures an edge-decomposition of an r-partite graph into r-cliques (subject to
trivially necessary divisibility conditions). The case of triangles translates
into the setting of partially completed Latin squares and more generally the
case of r-cliques translates into the setting of partially completed mutually
orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
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