389,970 research outputs found
Fundamental Cycles and Graph Embeddings
In this paper we present a new Good Characterization of maximum genus of a
graph which makes a common generalization of the works of Xuong, Liu, and Fu et
al. Based on this, we find a new polynomially bounded algorithm to find the
maximum genus of a graph
Graph cohomology and Kontsevich cycles
The dual Kontsevich cycles in the double dual of associative graph homology
correspond to polynomials in the Miller-Morita-Mumford classes in the integral
cohomology of mapping class groups. I explain how the coefficients of these
polynomials can be computed using Stasheff polyhedra and results from my
previous paper GT/0207042.Comment: 36 pages, 3 figure
A Message-Passing Algorithm for Counting Short Cycles in a Graph
A message-passing algorithm for counting short cycles in a graph is
presented. For bipartite graphs, which are of particular interest in coding,
the algorithm is capable of counting cycles of length g, g +2,..., 2g - 2,
where g is the girth of the graph. For a general (non-bipartite) graph, cycles
of length g; g + 1, ..., 2g - 1 can be counted. The algorithm is based on
performing integer additions and subtractions in the nodes of the graph and
passing extrinsic messages to adjacent nodes. The complexity of the proposed
algorithm grows as , where is the number of edges in the
graph. For sparse graphs, the proposed algorithm significantly outperforms the
existing algorithms in terms of computational complexity and memory
requirements.Comment: Submitted to IEEE Trans. Inform. Theory, April 21, 2010
P-matrices and signed digraphs
We associate a signed digraph with a list of matrices whose dimensions permit
them to be multiplied, and whose product is square. Cycles in this graph have a
parity, that is, they are either even (termed e-cycles) or odd (termed
o-cycles). The absence of e-cycles in the graph is shown to imply that the
matrix product is a P0-matrix, i.e., all of its principal minors are
nonnegative. Conversely, the presence of an e-cycle is shown to imply that
there exists a list of matrices associated with the graph whose product fails
to be a P0-matrix. The results generalise a number of previous results relating
P- and P0-matrices to graphs
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
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