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An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees. From a collection of subtrees of a tree, one can define a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed ...
In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs.
3. A strongly chordal graph is a chordal graph in which every cycle of length six or more has an odd chord. 4. A chordal bipartite graph is not chordal (unless it is a forest); it is a bipartite graph in which every cycle of six or more vertices has a chord, so the only induced cycles are 4-cycles. 5.
A line graph representing the change between different phases of matter, typically from a gas to a solid or a liquid to a solid, as a function of time and temperature; e.g. showing how the temperature of a liquid substance changes over time as it condenses below its freezing point. coordinate chemistry coordinate covalent bond See dipolar bond.
The split graphs are exactly the graphs that are chordal and have a chordal complement. [38] The k-trees, central to the definition of treewidth, are chordal graphs formed by starting with a (k + 1)-vertex clique and repeatedly adding a vertex so that it and its neighbors form a clique of the same size. [35]
In graph theory, a branch of mathematics, a chordal completion of a given undirected graph G is a chordal graph, on the same vertex set, that has G as a subgraph. A minimal chordal completion is a chordal completion such that any graph formed by removing an edge would no longer be a chordal completion. A minimum chordal completion is a chordal ...
Every perfectly orderable graph is a perfect graph. [1] Chordal graphs are perfectly orderable; a perfect ordering of a chordal graph may be found by reversing a perfect elimination ordering for the graph. Thus, applying greedy coloring to a perfect ordering provides an efficient algorithm for optimally coloring chordal graphs.
Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of star graphs. [5] Almost all chordal graphs are split graphs; that is, in the limit as n goes to infinity, the fraction of n-vertex chordal graphs that are split approaches one. [6] Because chordal graphs are ...