graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a ''k''-tree is an

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

formed by starting with a (''k'' + 1)-vertex

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices ...

and then repeatedly adding vertices in such a way that each added vertex ''v'' has exactly ''k'' neighbors ''U'' such that, together, the ''k'' + 1 vertices formed by ''v'' and ''U'' form a

clique A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popula ...

Characterizations

The ''k''-trees are exactly the maximal graphs with a

treewidth In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests. The gr ...

of ''k'' ("maximal" means that no more edges can be added without increasing their treewidth). They are also exactly the

chordal graph In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced c ...

s all of whose maximal cliques are the same size ''k'' + 1 and all of whose minimal clique separators are also all the same size ''k''..

Related graph classes

1-trees are the same as unrooted

trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

. 2-trees are maximal series–parallel graphs, and include also the maximal

outerplanar graph In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two fo ...

s. Planar 3-trees are also known as Apollonian networks. The graphs that have treewidth at most ''k'' are exactly the subgraphs of ''k''-trees, and for this reason they are called partial ''k''-trees.. The graphs formed by the edges and vertices of ''k''-dimensional

stacked polytope In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.. Examples Every simplex is itself a stacked polytope. In three dimensions, e ...

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s formed by starting from a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...

and then repeatedly gluing simplices onto the faces of the polytope, are ''k''-trees when ''k'' ≥ 3. This gluing process mimics the construction of ''k''-trees by adding vertices to a clique. A ''k''-tree is the graph of a stacked polytope if and only if no three (''k'' + 1)-vertex cliques have ''k'' vertices in common.

References

{{reflist Graph minor theory Trees (graph theory) Perfect graphs Graph families