1,024 research outputs found
Lift-and-project ranks of the stable set polytope of joined a-perfect graphs
In this paper we study lift-and-project polyhedral operators defined by
Lov?asz and Schrijver and Balas, Ceria and Cornu?ejols on the clique relaxation
of the stable set polytope of web graphs. We compute the disjunctive rank of
all webs and consequently of antiweb graphs. We also obtain the disjunctive
rank of the antiweb constraints for which the complexity of the separation
problem is still unknown. Finally, we use our results to provide bounds of the
disjunctive rank of larger classes of graphs as joined a-perfect graphs, where
near-bipartite graphs belong
Motif Clustering and Overlapping Clustering for Social Network Analysis
Motivated by applications in social network community analysis, we introduce
a new clustering paradigm termed motif clustering. Unlike classical clustering,
motif clustering aims to minimize the number of clustering errors associated
with both edges and certain higher order graph structures (motifs) that
represent "atomic units" of social organizations. Our contributions are
two-fold: We first introduce motif correlation clustering, in which the goal is
to agnostically partition the vertices of a weighted complete graph so that
certain predetermined "important" social subgraphs mostly lie within the same
cluster, while "less relevant" social subgraphs are allowed to lie across
clusters. We then proceed to introduce the notion of motif covers, in which the
goal is to cover the vertices of motifs via the smallest number of (near)
cliques in the graph. Motif cover algorithms provide a natural solution for
overlapping clustering and they also play an important role in latent feature
inference of networks. For both motif correlation clustering and its extension
introduced via the covering problem, we provide hardness results, algorithmic
solutions and community detection results for two well-studied social networks
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
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