3,981 research outputs found
K-Partite cliques of protein interactions: A novel subgraph topology for functional coherence analysis on PPI networks
Many studies are aimed at identifying dense clusters/subgraphs from protein-protein interaction (PPI) networks for protein function prediction. However, the prediction performance based on the dense clusters is actually worse than a simple guilt-by-association method using neighbor counting ideas. This indicates that the local topological structures and properties of PPI networks are still open to new theoretical investigation and empirical exploration. We introduce a novel topological structure called k-partite cliques of protein interactions-a functionally coherent but not-necessarily dense subgraph topology in PPI networks-to study PPI networks. A k-partite protein clique is a maximal k-partite clique comprising two or more nonoverlapping protein subsets between any two of which full interactions are exhibited. In the detection of PPI's maximal k-partite cliques, we propose to transform PPI networks into induced K-partite graphs where edges exist only between the partites. Then, we present a maximal k-partite clique mining (MaCMik) algorithm to enumerate maximal k-partite cliques from K-partite graphs. Our MaCMik algorithm is then applied to a yeast PPI network. We observed interesting and unusually high functional coherence in k-partite protein cliques-the majority of the proteins in k-partite protein cliques, especially those in the same partites, share the same functions, although k-partite protein cliques are not restricted to be dense compared with dense subgraph patterns or (quasi-)cliques. The idea of k-partite protein cliques provides a novel approach of characterizing PPI networks, and so it will help function prediction for unknown proteins.© 2013 Elsevier Ltd
Asymptotic multipartite version of the Alon-Yuster theorem
In this paper, we prove the asymptotic multipartite version of the
Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi
theorem: If is an integer, is a -colorable graph and
is fixed, then, for every sufficiently large , where
divides , and for every balanced -partite graph on vertices with
each of its corresponding bipartite subgraphs having minimum
degree at least , has a subgraph consisting of
vertex-disjoint copies of .
The proof uses the Regularity method together with linear programming.Comment: 22 pages, 1 figur
Graph Saturation in Multipartite Graphs
Let be a fixed graph and let be a family of graphs. A
subgraph of is -saturated if no member of
is a subgraph of , but for any edge in , some element of
is a subgraph of . We let and
denote the maximum and minimum size of an
-saturated subgraph of , respectively. If no element of
is a subgraph of , then .
In this paper, for and we determine
, where is the complete balanced -partite
graph with partite sets of size . We also give several families of
constructions of -saturated subgraphs of for . Our results
and constructions provide an informative contrast to recent results on the
edge-density version of from [A. Bondy, J. Shen, S.
Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite
graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs
in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure
Imposing Hierarchy on a Graph
This paper investigates a way of imposing a hierarchy on a graph in order to explore relationships between elements of data. Imposing a hierarchy is equivalent to clustering. First a tree structure is imposed on the initial graph, then a k-partite structure is imposed on each previously obtained cluster. Imposing a tree exposes the hierarchical structure of the graph as well as providing an abstraction of the data. In this study three kinds of merge operations are considered and their composition is shown to yield a tree with a maximal number of vertices in which vertices in the tree are associated with disjoint connected subgraphs. These subgraphs are subsequently transformed into k-partite graphs using similar merge operations. These merges also ensure that the obtained tree is proper with respect to the hierarchy imposed on the data.
A detailed example of the techniqueâs application in exposing the structure of protein interaction networks is described. The example focuses on the MAPK cell signalling pathway. The merge operations help expose where signal regulation occurs within the pathway and from other signalling pathways within the cell
An extremal theorem in the hypercube
The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two
vertices are adjacent if they differ in exactly one coordinate. For any
subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a
subgraph of Q_n which does not contain a copy of H. We find a wide class of
subgraphs H, including all previously known examples, for which ex(Q_n, H) =
o(e(Q_n)). In particular, our method gives a unified approach to proving that
ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.Comment: 6 page
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of
well known classical results in extremal graph theory. Here we follow this
trend and investigate the structure of triangle-free subgraphs of with
high minimum degree. We prove that asymptotically almost surely each
triangle-free spanning subgraph of with minimum degree at least
is -close to bipartite,
and each spanning triangle-free subgraph of with minimum degree at
least is -close to
-partite for some . These are random graph analogues of a
result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218],
and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show
that our results are best possible up to a constant factor.Comment: 18 page
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
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