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Characterizing Atoms that Result from Decomposition by Clique Separators

100%
McKee T. A.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
587-594
EN
A graph is defined to be an atom if no minimal vertex separator induces a complete subgraph; thus, atoms are the graphs that are immune to clique separator decomposition. Atoms are characterized here in two ways: first using generalized vertex elimination schemes, and then as generalizations of 2-connected unichord-free graphs (the graphs in which every minimal vertex separator induces an edgeless subgraph).
2
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Requiring that Minimal Separators Induce Complete Multipartite Subgraphs

100%
McKee T. A.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 38
|
nr 1
263-273
EN
Complete multipartite graphs range from complete graphs (with every partite set a singleton) to edgeless graphs (with a unique partite set). Requiring minimal separators to all induce one or the other of these extremes characterizes, respectively, the classical chordal graphs and the emergent unichord-free graphs. New theorems characterize several subclasses of the graphs whose minimal separators induce complete multipartite subgraphs, in particular the graphs that are 2-clique sums of complete, cycle, wheel, and octahedron graphs.
3
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A New Characterization of Unichord-Free Graphs

100%
McKee T. A.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 4
765-771
EN
Unichord-free graphs are defined as having no cycle with a unique chord. They have appeared in several papers recently and are also characterized by minimal separators always inducing edgeless subgraphs (in contrast to characterizing chordal graphs by minimal separators always inducing complete subgraphs). A new characterization of unichord-free graphs corresponds to a suitable reformulation of the standard simplicial vertex characterization of chordal graphs.
4
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Pairs Of Edges As Chords And As Cut-Edges

100%
McKee T. A.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 4
673-681
EN
Several authors have studied the graphs for which every edge is a chord of a cycle; among 2-connected graphs, one characterization is that the deletion of one vertex never creates a cut-edge. Two new results: among 3-connected graphs with minimum degree at least 4, every two adjacent edges are chords of a common cycle if and only if deleting two vertices never creates two adjacent cut-edges; among 4-connected graphs, every two edges are always chords of a common cycle.
5
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Maxclique and Unit Disk Characterizations of Strongly Chordal Graphs

64%
Caria P. D., McKee T. A.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 3
593-602
EN
Maxcliques (maximal complete subgraphs) and unit disks (closed neighborhoods of vertices) sometime play almost interchangeable roles in graph theory. For instance, interchanging them makes two existing characterizations of chordal graphs into two new characterizations. More intriguingly, these characterizations of chordal graphs can be naturally strengthened to new characterizations of strongly chordal graphs
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