Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
The topology of the independence complex
European Journal of Combinatorics
Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers
Journal of Algebraic Combinatorics: An International Journal
Shellable graphs and sequentially Cohen--Macaulay bipartite graphs
Journal of Combinatorial Theory Series A
Graph Theory
Note: Combinatorial Alexander Duality—A Short and Elementary Proof
Discrete & Computational Geometry
Certain Homology Cycles of the Independence Complex of Grids
Discrete & Computational Geometry
The tidy set: a minimal simplicial set for computing homology of clique complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
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A cross-cycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a cross-polytope and that contains a maximal face of K. A cross-cycle is an efficient way to define a non-zero class in the homology of K. For an independence complex of a graph G, a cross-cycle is equivalent to a combinatorial object: induced matching containing a maximal independent set. We study the complexity of finding cross-cycles in independence complexes. We show that in general this problem is NP-complete when input is a graph whose independence complex we consider. We then focus on the class of chordal graphs, where, as we show, cross-cycles detect all of homology of the independence complex. As our main result, we present a polynomial time algorithm for detecting a cross-cycle in the independence complex of a chordal graph. Our algorithm is based on the geometric intersection representation of chordal graphs and has an efficient implementation. As a corollary, we obtain polynomial time algorithms for such topological properties as contractibility or simple-connectedness of independence complexes of chordal graphs. These problems are undecidable for general independence complexes. We further prove that even for chordal graphs, it is NP-complete to decide if there is a cross-cycle of a given cardinality, and hence, if a particular homology group of the independence complex is nontrivial. As a corollary we obtain that computing homology groups of arbitrary simplicial complexes given as a list of facets is NP-hard.