# Forbidden graph characterization

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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).

## Definition

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 ${\displaystyle {\mathcal {F}}}$ if and only if a forbidden substructure is not contained in G. The forbidden substructure might be one of:

• subgraphs, smaller graphs obtained from subsets of the vertices and edges of a larger graph,
• induced subgraphs, smaller graphs obtained by selecting a subset of the vertices and using all edges with both endpoints in that subset,
• homeomorphic subgraphs (also called topological minors), smaller graphs obtained from subgraphs by collapsing paths of degree-two vertices to single edges, or
• graph minors, smaller graphs obtained from subgraphs by arbitrary edge contractions.

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.

## List of forbidden characterizations for graphs and hypergraphs

FamilyObstructionsRelationReference
Forests loops, pairs of parallel edges, and cycles of all lengthssubgraphDefinition
a loop (for multigraphs) or a triangle K3 (for simple graphs)graph minorDefinition
Claw-free graphs star K1,3induced subgraphDefinition
Comparability graphs induced subgraph
Triangle-free graphs triangle K3induced subgraphDefinition
Planar graphs K5 and K3,3homeomorphic subgraph Kuratowski's theorem
K5 and K3,3graph minor Wagner's theorem
Outerplanar graphs K4 and K2,3graph minor Diestel (2000), [1] p. 107
Outer 1-planar graphs six forbidden minorsgraph minor Auer et al. (2013) [2]
Graphs of fixed genus a finite obstruction setgraph minor Diestel (2000), [1] p. 275
Apex graphs a finite obstruction setgraph minor [3]
Linklessly embeddable graphs The Petersen family graph minor [4]
Bipartite graphs odd cyclessubgraph [5]
Chordal graphs cycles of length 4 or moreinduced subgraph [6]
Perfect graphs cycles of odd length 5 or more or their complements induced subgraph [7]
Line graph of graphs nine forbidden subgraphs (listed here)induced subgraph [8]
Graph unions of cactus graphs the four-vertex diamond graph formed by removing an edge from the complete graph K4graph minor [9]
Ladder graphs K2,3 and its dual graph homeomorphic subgraph [10]
Split graphs ${\displaystyle C_{4},C_{5},{\bar {C}}_{4}\left(=K_{2}+K_{2}\right)}$induced subgraph [11]
2-connected series-parallel (treewidth 2, branchwidth   2)K4graph minor Diestel (2000), [1] p. 327
Treewidth 3K5, octahedron, pentagonal prism, Wagner graph graph minor [12]
Branchwidth 3K5, octahedron, cube, Wagner graph graph minor [13]
Complement-reducible graphs (cographs) 4-vertex path P4induced subgraph [14]
Trivially perfect graphs 4-vertex path P4 and 4-vertex cycle C4induced subgraph [15]
Threshold graphs 4-vertex path P4, 4-vertex cycle C4, and complement of C4induced subgraph [15]
Line graph of 3-uniform linear hypergraphs a finite list of forbidden induced subgraphs with minimum degree at least 19induced 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 + 1induced subgraph [17] [18]
Graphs ΔY-reducible to a single vertexa finite list of at least 68 billion distinct (1,2,3)-clique sumsgraph minor [19]
General theorems
A family defined by an induced-hereditary property a, possibly non-finite, obstruction setinduced subgraph
A family defined by a minor-hereditary property a finite obstruction setgraph minor Robertson–Seymour theorem

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