TOPOLOGICAL PROPERTIES OF GRAPHICAL ARRANGEMENTS OF CHORDAL GRAPHS
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Abstract
We use the lexicographic shellability method to show that if G is a chordal graph, then the proper part of the intersection poset of the corresponding graphical arrangement AG has the homotopy type of a wedge of spheres. Furthermore, the number of spheres in the wedge is also indicated based on the characteristics of the graph G. Since a chordal graph is a supersolvable graph, it has a supersovable composition series of induced subgraphs. Each subgraph is corresponding to a modular element in the intersection poset of . We can find the number of decreasing maximal chains based on modular elements, or induced subgraphs in the composition series. The number of decreasing maximal chains is equal to the number of spheres in the wedge. The study of graphical arrangements of chordal graphs is an extension of other studies of graphical arrangements of complete graphs , or the Braid arrangement.
TOPOLOGICAL PROPERTIES OF GRAPHICAL ARRANGEMENTS OF CHORDAL GRAPHSNguyen Anh ThiUniversity of Science, Vietnam National University Ho Chi Minh City, VietnamCorresponding author: Nguyen Anh Thi – Email: nathi@hcmus.edu.vnReceived: September 17, 2020; Revised: December 24, 2020; Accepted: December 28, 2020
ABSTRACTWe use the lexicographic shellability method to show that if G is a chordal graph, then the proper part of the intersection poset of the corresponding graphical arrangement AG has the homotopy type of a wedge of spheres. Furthermore, the number of spheres in the wedge is also indicated based on the characteristics of the graph G. Since a chordal graph is a supersolvable graph, it has a supersovable composition series of induced subgraphs. Each subgraph is corresponding to a modular element in the intersection poset of . We can find the number of decreasing maximal chains based on modular elements, or induced subgraphs in the composition series. The number of decreasing maximal chains is equal to the number of spheres in the wedge. The study of graphical arrangements of chordal graphs is an extension of other studies of graphical arrangements of complete graphs , or the Braid arrangement.Keywords: chordal graphs; graphical arrangements; lexicographic shellability
Keywords
chordal graphs, graphical arrangements, lexicographic shellability
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References
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