2,940 research outputs found
On simplicial and co-simplicial vertices in graphs
AbstractWe investigate the class of graphs defined by the property that every induced subgraph has a vertex which is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set). In particular we give the list of minimal forbidden subgraphs for the subclass of graphs whose vertex-set can be emptied out by first recursively eliminating simplicial vertices and then recursively eliminating co-simplicial vertices
Kernel Bounds for Structural Parameterizations of Pathwidth
Assuming the AND-distillation conjecture, the Pathwidth problem of
determining whether a given graph G has pathwidth at most k admits no
polynomial kernelization with respect to k. The present work studies the
existence of polynomial kernels for Pathwidth with respect to other,
structural, parameters. Our main result is that, unless NP is in coNP/poly,
Pathwidth admits no polynomial kernelization even when parameterized by the
vertex deletion distance to a clique, by giving a cross-composition from
Cutwidth. The cross-composition works also for Treewidth, improving over
previous lower bounds by the present authors. For Pathwidth, our result rules
out polynomial kernels with respect to the distance to various classes of
polynomial-time solvable inputs, like interval or cluster graphs. This leads to
the question whether there are nontrivial structural parameters for which
Pathwidth does admit a polynomial kernelization. To answer this, we give a
collection of graph reduction rules that are safe for Pathwidth. We analyze the
success of these results and obtain polynomial kernelizations with respect to
the following parameters: the size of a vertex cover of the graph, the vertex
deletion distance to a graph where each connected component is a star, and the
vertex deletion distance to a graph where each connected component has at most
c vertices.Comment: This paper contains the proofs omitted from the extended abstract
published in the proceedings of Algorithm Theory - SWAT 2012 - 13th
Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201
Pure simplicial complexes and well-covered graphs
A graph is called well-covered if all maximal independent sets of
vertices have the same cardinality. A simplicial complex is called
pure if all of its facets have the same cardinality. Let be the
class of graphs with some disjoint maximal cliques covering all vertices. In
this paper, we prove that for any simplicial complex or any graph, there is a
corresponding graph in class with the same well-coveredness
property. Then some necessary and sufficient conditions are presented to
recognize fast when a graph in the class is well-covered or not. To do
this characterization, we use an algebraic interpretation according to
zero-divisor elements of the edge rings of graphs.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1009.524
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