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 at least 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 to exist 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.
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Family | Obstructions | Relation | Reference |
---|---|---|---|
Forests | Loops, pairs of parallel edges, and cycles of all lengths | Subgraph | Definition |
A loop (for multigraphs) or triangle K3 (for simple graphs) | Graph minor | Definition | |
Linear forests | [A loop / triangle K3 (see above)] and star K1,3 | 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), [1] p. 107 |
Outer 1-planar graphs | Six forbidden minors | Graph minor | Auer et al. (2013) [2] |
Graphs of fixed genus | A finite obstruction set | Graph minor | Diestel (2000), [1] p. 275 |
Apex graphs | A finite obstruction set | Graph minor | [3] |
Linklessly embeddable graphs | The Petersen family | Graph minor | [4] |
Bipartite graphs | Odd cycles | Subgraph | [5] |
Chordal graphs | Cycles of length 4 or more | Induced subgraph | [6] |
Perfect graphs | Cycles of odd length 5 or more or their complements | Induced subgraph | [7] |
Line graph of graphs | 9 forbidden subgraphs | Induced subgraph | [8] |
Graph unions of cactus graphs | The four-vertex diamond graph formed by removing an edge from the complete graph K4 | Graph minor | [9] |
Ladder graphs | K2,3 and its dual graph | Homeomorphic subgraph | [10] |
Split graphs | Induced subgraph | [11] | |
2-connected series–parallel (treewidth ≤ 2, branchwidth ≤ 2) | K4 | Graph minor | Diestel (2000), [1] p. 327 |
Treewidth ≤ 3 | K5, octahedron, pentagonal prism, Wagner graph | Graph minor | [12] |
Branchwidth ≤ 3 | K5, octahedron, cube, Wagner graph | Graph minor | [13] |
Complement-reducible graphs (cographs) | 4-vertex path P4 | Induced subgraph | [14] |
Trivially perfect graphs | 4-vertex path P4 and 4-vertex cycle C4 | Induced subgraph | [15] |
Threshold graphs | 4-vertex path P4, 4-vertex cycle C4, and complement of C4 | Induced subgraph | [15] |
Line graph of 3-uniform linear hypergraphs | A finite list of forbidden induced subgraphs with minimum degree at least 19 | Induced subgraph | [16] |
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 | [17] [18] |
Graphs ΔY-reducible to a single vertex | A finite list of at least 68 billion distinct (1,2,3)-clique sums | Graph minor | [19] |
Graphs of spectral radius at most | A finite obstruction set exists if and only if and for any , where is the largest root of . | Subgraph / induced subgraph | [20] |
Cluster graphs | three-vertex path graph | Induced subgraph | |
General theorems | |||
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 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 or of .
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.
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.
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.
In the mathematical area of graph theory, a clique is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph is an induced subgraph of that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.
In mathematics, a universal graph is an infinite graph that contains every finite graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family F: that is, an infinite graph belonging to F that contains all finite graphs in F. For instance, the Henson graphs are universal in this sense for the i-clique-free graphs.
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context of the problem.
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 nor K3,3. This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem.
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 graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly k are called k-trees, and the graphs with treewidth at most k are called partial k-trees. Many other well-studied graph families also have bounded treewidth.
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number that measures how much the path was thickened to form G. More formally, a path-decomposition is a sequence of subsets of vertices of G such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness, vertex separation number, or node searching number.
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In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross, and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.
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