Abstract
The 2-connected 2-tree graphs are defined as being constructible from a single 3-cycle by recursively appending new degree-2 vertices so as to form 3-cycles that have unique edges in common with the existing graph. Such 2-trees can be characterized both as the edge-minimal chordal graphs and also as the edge-maximal series-parallel graphs. These are also precisely the 2-connected graphs that are simultaneously chordal and series-parallel, where these latter two better-known types of graphs have themselves been both characterized and applied in numerous ways that are unmotivated by their interaction with 2-trees and with each other.
Highlights
Introduction to the relevant graph classesThe 2-dimensional trees—commonly called 2-trees—can be defined recursively by beginning with a single edge and repeatedly appending new degree-2 vertices, each with two new edges so as to form a new 3-cycle—a K3 triangle—with existing edges of previously constructed 2-trees; see [12]
The present paper examines several simple, very closely-related characterizations of chordal graphs and 2-trees and, after that, of seriesparallel graphs and 2-trees
This leads to showing a way in which chordal graphs and series-parallel graphs are special—extremal—in this regard
Summary
Introduction to the relevant graph classesThe 2-dimensional trees—commonly called 2-trees—can be defined recursively by beginning with a single edge and repeatedly appending new degree-2 vertices, each with two new edges so as to form a new 3-cycle—a K3 triangle—with existing edges of previously constructed 2-trees; see [12]. This enables the following characterization to be proved from the two propositions above in [8]: The nontrivial 2-trees are the 2-connected graphs that are simultaneously chordal and series-parallel . By Proposition 1.2, inserting a new edge st into G will create a 2-connected graph G+ that is not series-parallel.
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