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).

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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 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.

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 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

See also

Related Research Articles

Graph theory Area of discrete mathematics

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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.

Kuratowskis 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 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.

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.

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 and vertices and by contracting edges.

Clique (graph theory)

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, its induced subgraph 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 the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).

Snark (graph theory)

In the mathematical field of graph theory, a snark is a simple, connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, the connectivity is redundant so that removing no one edge would split the graph, and the edges cannot be colored by only three colors without two edges of the same color meeting at a point. In order to avoid trivial cases, snarks are often restricted to have girth at least 5.

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Wagners theorem Characterization theorem in graph theory of planar graphs

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Branch-decomposition

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Apollonian network

In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

Planar cover

In graph theory, a planar cover of a finite graph G is a finite covering graph of G that is itself a planar graph. Every graph that can be embedded into the projective plane has a planar cover; an unsolved conjecture of Seiya Negami states that these are the only graphs with planar covers.

In graph theory, a family of graphs is said to have bounded expansion if all of its shallow minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion, is equivalent to the existence of separator theorems for these families. Families with these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs.

References

  1. 1 2 3 Diestel, Reinhard (2000), Graph Theory, Graduate Texts in Mathematics, 173, Springer-Verlag, ISBN   0-387-98976-5 .
  2. Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Gleißner, Andreas; Hanauer, Kathrin; Neuwirth, Daniel; Reislhuber, Josef (2013), "Recognizing outer 1-planar graphs in linear time", in Wismath, Stephen; Wolff, Alexander (eds.), 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, Lecture Notes in Computer Science, 8242, pp. 107–118, doi: 10.1007/978-3-319-03841-4_10 .
  3. Gupta, A.; Impagliazzo, R. (1991), "Computing planar intertwines", Proc. 32nd IEEE Symposium on Foundations of Computer Science (FOCS '91) , IEEE Computer Society, pp. 802–811, doi:10.1109/SFCS.1991.185452 .
  4. Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993), "Linkless embeddings of graphs in 3-space", Bulletin of the American Mathematical Society, 28 (1): 84–89, arXiv: math/9301216 , doi:10.1090/S0273-0979-1993-00335-5, MR   1164063 .
  5. Béla Bollobás (1998) "Modern Graph Theory", Springer, ISBN   0-387-98488-7 p. 9
  6. Kashiwabara, Toshinobu (1981), "Algorithms for some intersection graphs", in Saito, Nobuji; Nishizeki, Takao (eds.), Graph Theory and Algorithms, 17th Symposium of Research Institute of Electric Communication, Tohoku University, Sendai, Japan, October 24-25, 1980, Proceedings, Lecture Notes in Computer Science, 108, Springer-Verlag, pp. 171–181, doi:10.1007/3-540-10704-5\_15 .
  7. Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem" (PDF), Annals of Mathematics , 164 (1): 51–229, arXiv: math/0212070v1 , doi:10.4007/annals.2006.164.51 .
  8. Beineke, L. W. (1968), "Derived graphs of digraphs", in Sachs, H.; Voss, H.-J.; Walter, H.-J. (eds.), Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33.
  9. El-Mallah, Ehab; Colbourn, Charles J. (1988), "The complexity of some edge deletion problems", IEEE Transactions on Circuits and Systems, 35 (3): 354–362, doi:10.1109/31.1748 .
  10. Takamizawa, K.; Nishizeki, Takao; Saito, Nobuji (1981), "Combinatorial problems on series-parallel graphs", Discrete Applied Mathematics, 3 (1): 75–76, doi:10.1016/0166-218X(81)90031-7 .
  11. Földes, Stéphane; Hammer, Peter Ladislaw (1977a), "Split graphs", Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), Congressus Numerantium, XIX, Winnipeg: Utilitas Math., pp. 311–315, MR   0505860
  12. Bodlaender, Hans L. (1998), "A partial k-arboretum of graphs with bounded treewidth", Theoretical Computer Science, 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4 .
  13. Bodlaender, Hans L.; Thilikos, Dimitrios M. (1999), "Graphs with branchwidth at most three", Journal of Algorithms, 32 (2): 167–194, doi:10.1006/jagm.1999.1011 .
  14. Seinsche, D. (1974), "On a property of the class of n-colorable graphs", Journal of Combinatorial Theory, Series B, 16 (2): 191–193, doi:10.1016/0095-8956(74)90063-X, MR   0337679
  15. 1 2 Golumbic, Martin Charles (1978), "Trivially perfect graphs", Discrete Mathematics, 24 (1): 105–107, doi:10.1016/0012-365X(78)90178-4 ..
  16. Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of linear 3-uniform hypergraphs", Journal of Graph Theory, 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR   1459889
  17. Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), "Recognizing intersection graphs of linear uniform hypergraphs", Graphs and Combinatorics , 13: 359–367, doi:10.1007/BF03353014, MR   1485929
  18. Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1982), "Intersection graphs of k-uniform hypergraphs", European Journal of Combinatorics, 3: 159–172, doi:10.1016/s0195-6698(82)80029-2, MR   0670849
  19. Yu, Yanming (2006), "More Forbidden Minors for Wye-Delta-Wye Reducibility", The Electronic Journal of Combinatorics, 13 Website