In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of one of these two graphs as a subgraph (in which case it does not belong to the planar graphs).
More generally, a forbidden graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden from existing within any graph in the family. Different families vary in the nature of what is forbidden. In general, a structure G is a member of a family if and only if a forbidden substructure is not contained in G. The forbidden substructure might be one of:
The set of structures that are forbidden from belonging to a given graph family can also be called an obstruction set for that family.
Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
In order for a family to have a forbidden graph characterization, with a particular type of substructure, the family must be closed under substructures. That is, every substructure (of a given type) of a graph in the family must be another graph in the family. Equivalently, if a graph is not part of the family, all larger graphs containing it as a substructure must also be excluded from the family. When this is true, there always exists an obstruction set (the set of graphs that are not in the family but whose smaller substructures all belong to the family). However, for some notions of what a substructure is, this obstruction set could be infinite. The Robertson–Seymour theorem proves that, for the particular case of graph minors, a family that is closed under minors always has a finite obstruction set.
|Forests||loops, pairs of parallel edges, and cycles of all lengths||subgraph||Definition|
|a loop (for multigraphs) or a triangle K3 (for simple graphs)||graph minor||Definition|
|Claw-free graphs||star K1,3||induced subgraph||Definition|
|Comparability graphs||induced subgraph|
|Triangle-free graphs||triangle K3||induced subgraph||Definition|
|Planar graphs||K5 and K3,3||homeomorphic subgraph||Kuratowski's theorem|
|K5 and K3,3||graph minor||Wagner's theorem|
|Outerplanar graphs||K4 and K2,3||graph minor||Diestel (2000), p. 107|
|Outer 1-planar graphs||six forbidden minors||graph minor||Auer et al. (2013)|
|Graphs of fixed genus||a finite obstruction set||graph minor||Diestel (2000), p. 275|
|Apex graphs||a finite obstruction set||graph minor|
|Linklessly embeddable graphs||The Petersen family||graph minor|
|Bipartite graphs||odd cycles||subgraph|
|Chordal graphs||cycles of length 4 or more||induced subgraph|
|Perfect graphs||cycles of odd length 5 or more or their complements||induced subgraph|
|Line graph of graphs||nine forbidden subgraphs (listed here)||induced subgraph|
|Graph unions of cactus graphs||the four-vertex diamond graph formed by removing an edge from the complete graph K4||graph minor|
|Ladder graphs||K2,3 and its dual graph||homeomorphic subgraph|
|Split graphs||induced subgraph|
|2-connected series-parallel (treewidth ≤ 2, branchwidth ≤ 2)||K4||graph minor||Diestel (2000), p. 327|
|Treewidth ≤ 3||K5, octahedron, pentagonal prism, Wagner graph||graph minor|
|Branchwidth ≤ 3||K5, octahedron, cube, Wagner graph||graph minor|
|Complement-reducible graphs (cographs)||4-vertex path P4||induced subgraph|
|Trivially perfect graphs||4-vertex path P4 and 4-vertex cycle C4||induced subgraph|
|Threshold graphs||4-vertex path P4, 4-vertex cycle C4, and complement of C4||induced subgraph|
|Line graph of 3-uniform linear hypergraphs||a finite list of forbidden induced subgraphs with minimum degree at least 19||induced subgraph|
|Line graph of k-uniform linear hypergraphs, k > 3||a finite list of forbidden induced subgraphs with minimum edge degree at least 2k2 − 3k + 1||induced subgraph|
|Graphs ΔY-reducible to a single vertex||a finite list of at least 68 billion distinct (1,2,3)-clique sums||graph minor|
|A family defined by an induced-hereditary property||a, possibly non-finite, obstruction set||induced subgraph|
|A family defined by a minor-hereditary property||a finite obstruction set||graph minor||Robertson–Seymour theorem|
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 which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 or of K3,3.
In graph theory, the Robertson–Seymour theorem states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K5 or the complete bipartite graph K3,3 as minors.
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