In 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 *vertices* (also called *nodes* or *points*) and each of the related pairs of vertices is called an *edge* (also called *link* or *line*).^{[1]} Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person *A* can shake hands with a person *B* only if *B* also shakes hands with *A*. In contrast, if any edge from a person *A* to a person *B* corresponds to *A* admiring *B*, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called an **undirected graph** while the latter type of graph is called a **directed graph**.

Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by James Joseph Sylvester in 1878.^{[2]}^{[3]}

Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.

A **graph** (sometimes called *undirected graph* for distinguishing to from a directed graph, or *simple graph* for distinguishing from a multigraph)^{[4]}^{[5]} is a pair *G* = (*V*, *E*), where V is a set whose elements are called *vertices* (singular: vertex), and E is a set of two-sets (set with two distinct elements) of vertices, whose elements are called *edges* (sometimes *links* or *lines*).

The vertices *x* and *y* of an edge {*x*, *y*} are called the *endpoints* of the edge. The edge is said to *join* *x* and *y* and to be *incident* on *x* and *y*. A vertex may not belong to any edge.

A multigraph is a generalization that allows multiple edges adjacent to the same pair of vertices. In some texts, multigraphs are simply called graphs.^{[6]}^{[7]}

Sometimes, graphs are allowed to contain *loops*, which are edges that join a vertex to itself. For allowing loops, the above definition must be changed by defining edges as multisets of two vertices instead of two-sets. Such generalized graphs are called *graphs with loops* or simply *graphs* when it is clear from the context that loops are allowed.

Generally, the set of vertices *V* is supposed to be finite; this implies that the set of edges is also finite. Infinite graphs are sometimes considered, but are more often viewed as a special kind of binary relation, as most results on finite graphs do not extend to the infinite case, or need a rather different proof.

An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The *order* of a graph is its number of vertices |*V*|. The *size* of a graph is its number of edges |*E*|. However, in some contexts, such that for expressing the computational complexity of algorithms, the size is |*V*| + |*E*| (otherwise, a non-empty graph could have a size 0). The *degree* or *valency* of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice.

In a graph of order *n*, the maximum degree of each vertex is *n* − 1 (or *n* + 1 if loops are allowed), and the maximum number of edges is *n*(*n* − 1)/2 (or *n*(*n* + 1)/2 if loops are allowed).

The edges of a graph define a symmetric relation on the vertices, called the *adjacency relation*. Specifically, two vertices *x* and *y* are *adjacent* if {*x*, *y*} is an edge.

A **directed graph** or **digraph** is a graph in which edges have orientations.

In one restricted but very common sense of the term,^{[8]} a **directed graph** is an ordered pair *G* = (*V*, *E*) comprising:

*V*a set of*vertices*(also called*nodes*or*points*);*E*⊆ {(*x*,*y*) | (*x*,*y*) ∈*V*^{2}∧*x*≠*y*} a set of*edges*(also called*directed edges*,*directed links*,*directed lines*,*arrows*or*arcs*) which are ordered pairs of*distinct*vertices (i.e., an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely a **directed simple graph**.

In the edge (*x*, *y*) directed from *x* to *y*, the vertices *x* and *y* are called the *endpoints* of the edge, *x* the *tail* of the edge and *y* the *head* of the edge. The edge (*y*, *x*) is called the *inverted edge* of (*x*, *y*). The edge is said to *join* *x* and *y* and to be *incident* on *x* and on *y*. A vertex may exist in a graph and not belong to an edge. A *loop* is an edge that joins a vertex to itself. *Multiple edges* are two or more edges that join the same two vertices.

In one more general sense of the term allowing multiple edges,^{[8]} a **directed graph** is an ordered triple *G* = (*V*, *E*, *ϕ*) comprising:

*V*a set of*vertices*(also called*nodes*or*points*);*E*a set of*edges*(also called*directed edges*,*directed links*,*directed lines*,*arrows*or*arcs*);*ϕ*:*E*→ {(*x*,*y*) | (*x*,*y*) ∈*V*^{2}∧ x ≠ y} an*incidence function*mapping every edge to an ordered pair of*distinct*vertices (i.e., an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely a **directed multigraph**.

Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex *x* is the edge (for a directed simple graph) or is incident on (for a directed multigraph) (*x*, *x*) which is not in {(*x*, *y*) | (*x*, *y*) ∈ *V*^{2} ∧ *x* ≠ *y*}. So to allow loops the definitions must be expanded. For directed simple graphs, *E* ⊆ {(*x*, *y*) | (*x*, *y*) ∈ *V*^{2} ∧ x ≠ y} should become *E* ⊆ *V*^{2}. For directed multigraphs, *ϕ*: *E* → {(*x*, *y*) | (*x*, *y*) ∈ *V*^{2} ∧ x ≠ y} should become *ϕ*: *E* → *V*^{2}. To avoid ambiguity, these types of objects may be called precisely a **directed simple graph permitting loops** and a **directed multigraph permitting loops** (or a *quiver*) respectively.

The edges of a directed simple graph permitting loops *G* is a homogeneous relation ~ on the vertices of *G* that is called the *adjacency relation* of *G*. Specifically, for each edge (*x*, *y*), its endpoints *x* and *y* are said to be *adjacent* to one another, which is denoted *x* ~ *y*.

A *mixed graph* is a graph in which some edges may be directed and some may be undirected. It is an ordered triple *G* = (*V*, *E*, *A*) for a *mixed simple graph* and *G* = (*V*, *E*, *A*, *ϕ*_{E}, *ϕ*_{A}) for a *mixed multigraph* with *V*, *E* (the undirected edges), *A* (the directed edges), *ϕ*_{E} and *ϕ*_{A} defined as above. Directed and undirected graphs are special cases.

A *weighted graph* or a *network*^{[9]}^{[10]} is a graph in which a number (the weight) is assigned to each edge.^{[11]} Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Such graphs arise in many contexts, for example in shortest path problems such as the traveling salesman problem.

An *oriented graph* is a directed graph in which at most one of (*x*, *y*) and (*y*, *x*) may be edges of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected graph. However, some authors use "oriented graph" to mean the same as "directed graph".

A *regular graph* is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree *k* is called a *k*‑regular graph or regular graph of degree *k*.

A *complete graph* is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.

A *finite graph* is a graph in which the vertex set and the edge set are finite sets. Otherwise, it is called an *infinite graph*.

Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.

In an undirected graph, an unordered pair of vertices {*x*, *y*} is called *connected* if a path leads from *x* to *y*. Otherwise, the unordered pair is called *disconnected*.

A *connected graph* is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a *disconnected graph*.

In a directed graph, an ordered pair of vertices (*x*, *y*) is called *strongly connected* if a directed path leads from *x* to *y*. Otherwise, the ordered pair is called *weakly connected* if an undirected path leads from *x* to *y* after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called *disconnected*.

A *strongly connected graph* is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a *weakly connected graph* if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a *disconnected graph*.

A *k-vertex-connected graph* or *k-edge-connected graph* is a graph in which no set of *k* − 1 vertices (respectively, edges) exists that, when removed, disconnects the graph. A *k*-vertex-connected graph is often called simply a *k-connected graph*.

A *bipartite graph* is a simple graph in which the vertex set can be partitioned into two sets, *W* and *X*, so that no two vertices in *W* share a common edge and no two vertices in *X* share a common edge. Alternatively, it is a graph with a chromatic number of 2.

In a complete bipartite graph, the vertex set is the union of two disjoint sets, *W* and *X*, so that every vertex in *W* is adjacent to every vertex in *X* but there are no edges within *W* or *X*.

A *path graph* or *linear graph* of order *n* ≥ 2 is a graph in which the vertices can be listed in an order *v*_{1}, *v*_{2}, …, *v*_{n} such that the edges are the {*v*_{i}, *v*_{i+1}} where *i* = 1, 2, …, *n* − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph.

A *planar graph* is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.

A *cycle graph* or *circular graph* of order *n* ≥ 3 is a graph in which the vertices can be listed in an order *v*_{1}, *v*_{2}, …, *v*_{n} such that the edges are the {*v*_{i}, *v*_{i+1}} where *i* = 1, 2, …, *n* − 1, plus the edge {*v*_{n}, *v*_{1}}. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.

A *tree* is an undirected graph in which any two vertices are connected by *exactly one* path, or equivalently a connected acyclic undirected graph.

A *forest* is an undirected graph in which any two vertices are connected by *at most one* path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.

A *polytree* (or *directed tree* or *oriented tree* or *singly connected network*) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree.

A *polyforest* (or *directed forest* or *oriented forest*) is a directed acyclic graph whose underlying undirected graph is a forest.

More advanced kinds of graphs are:

- Petersen graph and its generalizations;
- perfect graphs;
- cographs;
- chordal graphs;
- other graphs with large automorphism groups: vertex-transitive, arc-transitive, and distance-transitive graphs;
- strongly regular graphs and their generalizations distance-regular graphs.

Two edges of a graph are called *adjacent* if they share a common vertex. Two edges of a directed graph are called *consecutive* if the head of the first one is the tail of the second one. Similarly, two vertices are called *adjacent* if they share a common edge (*consecutive* if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to *join* the two vertices. An edge and a vertex on that edge are called *incident*.

The graph with only one vertex and no edges is called the *trivial graph*. A graph with only vertices and no edges is known as an *edgeless graph*. The graph with no vertices and no edges is sometimes called the *null graph* or *empty graph*, but the terminology is not consistent and not all mathematicians allow this object.

Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called *vertex-labeled*. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called *edge-labeled*. Graphs with labels attached to edges or vertices are more generally designated as *labeled*. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called *unlabeled*. (Note that in the literature, the term *labeled* may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)

The category of all graphs is the slice category Set ↓ *D* where *D*: Set → Set is the functor taking a set *s* to *s* × *s*.

- The diagram is a schematic representation of the graph with vertices and edges
- In computer science, directed graphs are used to represent knowledge (e.g., conceptual graph), finite state machines, and many other discrete structures.
- A binary relation
*R*on a set*X*defines a directed graph. An element*x*of*X*is a direct predecessor of an element*y*of*X*if and only if*xRy*. - A directed graph can model information networks such as Twitter, with one user following another.
^{[12]}^{[13]} - Particularly regular examples of directed graphs are given by the Cayley graphs of finitely-generated groups, as well as Schreier coset graphs
- In category theory, every small category has an underlying directed multigraph whose vertices are the objects of the category, and whose edges are the arrows of the category. In the language of category theory, one says that there is a forgetful functor from the category of small categories to the category of quivers.

There are several operations that produce new graphs from initial ones, which might be classified into the following categories:

*unary operations*, which create a new graph from an initial one, such as:*binary operations*, which create a new graph from two initial ones, such as:

In a hypergraph, an edge can join more than two vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

Every graph gives rise to a matroid.

In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number, see continuous graph.

In computational biology, power graph analysis introduces power graphs as an alternative representation of undirected graphs.

In geographic information systems, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids.

- Conceptual graph
- Dual graph
- Graph (abstract data type)
- Graph database
- Graph drawing
- List of graph theory topics
- List of publications in graph theory
- Network theory

**^**Trudeau, Richard J. (1993).*Introduction to Graph Theory*(Corrected, enlarged republication. ed.). New York: Dover Pub. p. 19. ISBN 978-0-486-67870-2. Retrieved 8 August 2012.A graph is an object consisting of two sets called its

*vertex set*and its*edge set*.**^**See:- J. J. Sylvester (February 7, 1878) "Chemistry and algebra,"
*Nature*,*17*: 284. doi:10.1038/017284a0. From page 284: "Every invariant and covariant thus becomes expressible by a*graph*precisely identical with a Kekuléan diagram or chemicograph." - J. J. Sylvester (1878) "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, – with three appendices,"
*American Journal of Mathematics, Pure and Applied*,*1*(1) : 64–90. doi:10.2307/2369436. JSTOR 2369436. The term "graph" first appears in this paper on page 65.

- J. J. Sylvester (February 7, 1878) "Chemistry and algebra,"
**^**Gross, Jonathan L.; Yellen, Jay (2004).*Handbook of graph theory*. CRC Press. p. 35. ISBN 978-1-58488-090-5.**^**Bender & Williamson 2010, p. 148.**^**See, for instance, Iyanaga and Kawada,*69 J*, p. 234 or Biggs, p. 4.**^**Bender & Williamson 2010, p. 149.**^**Graham et al., p. 5.- ^
^{a}^{b}Bender & Williamson 2010, p. 161. **^**Strang, Gilbert (2005),*Linear Algebra and Its Applications*(4th ed.), Brooks Cole, ISBN 978-0-03-010567-8**^**Lewis, John (2013),*Java Software Structures*(4th ed.), Pearson, p. 405, ISBN 978-0133250121**^**Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991).*Foundations of Discrete Mathematics*(International student ed.). Boston: PWS-KENT Pub. Co. p. 463. ISBN 978-0-53492-373-0.A

*weighted graph*is a graph in which a number*w(e)*, called its*weight*, is assigned to each edge*e*.**^**Grandjean, Martin (2016). "A social network analysis of Twitter: Mapping the digital humanities community".*Cogent Arts & Humanities*.**3**(1): 1171458. doi:10.1080/23311983.2016.1171458.**^**Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zadeh WTF: The who-to-follow system at Twitter,*Proceedings of the 22nd international conference on World Wide Web*. doi:10.1145/2488388.2488433.

- Balakrishnan, V. K. (1997).
*Graph Theory*(1st ed.). McGraw-Hill. ISBN 978-0-07-005489-9. - Bang-Jensen, J.; Gutin, G. (2000).
*Digraphs: Theory, Algorithms and Applications*. Springer. - Bender, Edward A.; Williamson, S. Gill (2010).
*Lists, Decisions and Graphs. With an Introduction to Probability*. - Berge, Claude (1958).
*Théorie des graphes et ses applications*(in French). Paris: Dunod. - Biggs, Norman (1993).
*Algebraic Graph Theory*(2nd ed.). Cambridge University Press. ISBN 978-0-521-45897-9. - Bollobás, Béla (2002).
*Modern Graph Theory*(1st ed.). Springer. ISBN 978-0-387-98488-9. - Diestel, Reinhard (2005).
*Graph Theory*(3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-26183-4. - Graham, R.L.; Grötschel, M.; Lovász, L. (1995).
*Handbook of Combinatorics*. MIT Press. ISBN 978-0-262-07169-7. - Gross, Jonathan L.; Yellen, Jay (1998).
*Graph Theory and Its Applications*. CRC Press. ISBN 978-0-8493-3982-0. - Gross, Jonathan L.; Yellen, Jay (2003).
*Handbook of Graph Theory*. CRC. ISBN 978-1-58488-090-5. - Harary, Frank (1995).
*Graph Theory*. Addison Wesley Publishing Company. ISBN 978-0-201-41033-4. - Iyanaga, Shôkichi; Kawada, Yukiyosi (1977).
*Encyclopedic Dictionary of Mathematics*. MIT Press. ISBN 978-0-262-09016-2. - Zwillinger, Daniel (2002).
*CRC Standard Mathematical Tables and Formulae*(31st ed.). Chapman & Hall/CRC. ISBN 978-1-58488-291-6.

- Trudeau, Richard J. (1993).
*Introduction to Graph Theory*(Corrected, enlarged republication. ed.). New York: Dover Publications. ISBN 978-0-486-67870-2. Retrieved 8 August 2012.

- Media related to Graph (discrete mathematics) at Wikimedia Commons
- Weisstein, Eric W. "Graph".
*MathWorld*.

In mathematics, the bondage number of a nonempty graph is the cardinality of the smallest set E of edges such that the domination number of the graph with the edges E removed is strictly greater than the domination number of the original graph.

The concept was introduced by Fink et. al.

Cluster graphIn graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs.

Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphs and the 2-leaf powers.

Connected dominating setIn graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.

Core (graph theory)In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.

GephiGephi is an open-source network analysis and visualization software package written in Java on the NetBeans platform.

Graph theoryIn 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 connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph (discrete mathematics) 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.

Refer to the glossary of graph theory for basic definitions in graph theory.

Incidence posetIn mathematics, an incidence poset or incidence order is a type of partially ordered set that represents the incidence relation between vertices and edges of an undirected graph. The incidence poset of a graph G has an element for each vertex or edge in G; in this poset, there is an order relation x ≤ y if and only if either x = y or x is a vertex, y is an edge, and x is an endpoint of y.

IncidentIncident may refer to:

A property of a graph (discrete mathematics) (see also glossary of graph theory)

Incident (film), a 1948 film noir

Incident (festival), a cultural festival of The National Institute of Technology in Surathkal, Karnataka, India

Incident (Scientology), a concept in Scientology

Incident Ray, a ray of light that strikes a surface

Nuclear and radiation accidents and incidents, an irregularity with a nuclear installation not classified as a nuclear accident

Klam valueIn the parameterized complexity of algorithms, the klam value of a parameterized algorithm is a number that bounds the parameter values for which the algorithm might reasonably be expected to be practical. An algorithm with a higher klam value can be used for a wider range of parameter values than another algorithm with a lower klam value. The klam value was first defined by Downey and Fellows (1999), and has since been used by other researchers in parameterized complexity both as a way of comparing different algorithms to each other and in order to set goals for future algorithmic improvements.

Logical matrixA logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets.

Meurs ChallengerMeurs Challenger is an online graph visualization program, with data analysis and browsing.The software supports several graph layout algorithms, and allows the user to interact with the nodes. The displayed data can be filtered using textual search, node and edge type, or based on the graph distance between nodes. Written in ActionScript, the program runs on Windows, Linux, macOS and other platforms that support the Adobe Flash Player.

Meurs Challenger was the winner at the 2011 edition of the International Symposium on Graph Drawing, in the large graph category.It is publicly available as a Facebook application, which displays the network graph of the user's friends.

Mind mapA mind map is a diagram used to visually organize information. A mind map is hierarchical and shows relationships among pieces of the whole. It is often created around a single concept, drawn as an image in the center of a blank page, to which associated representations of ideas such as images, words and parts of words are added. Major ideas are connected directly to the central concept, and other ideas branch out from those major ideas.

Mind maps can also be drawn by hand, either as "rough notes" during a lecture, meeting or planning session, for example, or as higher quality pictures when more time is available. Mind maps are considered to be a type of spider diagram. A similar concept in the 1970s was "idea sun bursting".

Order dimensionIn mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.

This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order.

Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).

Paul A. CatlinPaul Allen Catlin ((1948-06-25)June 25, 1948 – (1995-04-20)April 20, 1995) was a mathematician, professor of mathematics and Doctor of Mathematics, known for his valuable contributions to graph theory and number theory. He wrote one of the most cited papers in the series of chromatic numbers and Brooks' theorem, titled Hajós graph coloring conjecture: variations and counterexamples.

Pursuit-evasionPursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit-evasion, and the graph formulation discrete pursuit-evasion (also called graph searching). Current research is typically limited to one of these two formulations.

SubcoloringIn graph theory, a subcoloring is an assignment of colors to a graph's vertices such that each color class induces a vertex disjoint union of cliques. That is, each color class should form a cluster graph.

The subchromatic number χS(G) of a graph G is the least number of colors needed in any subcoloring of G.

Subcoloring and subchromatic number were introduced by Albertson et al. (1989).

Every proper coloring and cocoloring of a graph are also subcolorings, so the subchromatic number of any graph is at most equal to the cochromatic number, which is at most equal to the chromatic number.

Subcoloring is as difficult to solve exactly as coloring, in the sense that (like coloring) it is NP-complete. More specifically,

the problem of determining whether a planar graph has subchromatic number at most 2 is NP-complete, even if it is a

triangle-free graph with maximum degree 4 (Gimbel & Hartman 2003) (Fiala et al. 2003),

comparability graph with maximum degree 4 (Ochem 2017),

line graph of a bipartite graph with maximum degree 4 (Gonçalves & Ochem 2009),

graph with girth 5 (Montassier & Ochem 2015).The subchromatic number of a cograph can be computed in polynomial time (Fiala et al. 2003). For every fixed integer r, it is possible to decide in polynomial time whether the subchromatic number of interval and permutation graphs is at most r (Broersma et al. 2002).

Three-dimensional graphA three-dimensional graph may refer to

A graph (discrete mathematics), embedded into a three-dimensional space

The graph of a function of two variables, embedded into a three-dimensional space

Well-colored graphIn graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number (minimum number of colors) and Grundy number (maximum number of greedily-chosen colors) are equal.

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