291 research outputs found
A Dirac-type Characterization of k-chordal Graphs
Characterization of k-chordal graphs based on the existence of a "simplicial
path" was shown in [Chv{\'a}tal et al. Note: Dirac-type characterizations of
graphs without long chordless cycles. Discrete Mathematics, 256, 445-448,
2002]. We give a characterization of k-chordal graphs which is a generalization
of the known characterization of chordal graphs due to [G. A. Dirac. On rigid
circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25, 71-76, 1961] that use
notions of a "simplicial vertex" and a "simplicial ordering".Comment: 3 page
Vertex decomposable graphs and obstructions to shellability
Inspired by several recent papers on the edge ideal of a graph G, we study
the equivalent notion of the independence complex of G. Using the tool of
vertex decomposability from geometric combinatorics, we show that 5-chordal
graphs with no chordless 4-cycles are shellable and sequentially
Cohen-Macaulay. We use this result to characterize the obstructions to
shellability in flag complexes, extending work of Billera, Myers, and Wachs. We
also show how vertex decomposability may be used to show that certain graph
constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and
additional references. v3: minor corrections for publicatio
Cohen-Macaulay binomial edge ideals
We study the depth of classes of binomial edge ideals and classify all closed
graphs whose binomial edge ideal is Cohen--Macaulay.Comment: 9 page
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
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