Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K < sub > 4 </ sub > or the complete bipartite graph K < sub > 2, 3 </ sub >.

Outerplanar graphs have degeneracy at most two: every subgraph of an outerplanar graph contains a vertex with degree at most two.

Outerplanar graphs have treewidth at most two, which implies that many graph optimization problems that are NP-complete for arbitrary graphs may be solved in polynomial time by dynamic programming when the input is outerplanar.

* Outerplanar graphs, Information System on Graph Classes and Their Inclusions

If this were the restriction, planar graphs would require arbitrarily large numbers of colors.

Historically they were sometimes called Feynman-Dyson diagrams or Dyson graphs, because when they were introduced the path integral was unfamiliar, and Freeman Dyson's derivation from old-fashioned perturbation theory was easier to follow for physicists trained in earlier methods.

The European Commission had some difficulty funding the project's next stage, after several allegedly " per annum " sales projection graphs for the project were exposed in November 2001 as " cumulative " projections ( which for each year projected, necessarily included all previous years of sales ).

Although the episodes were scripted, most of the music used over the two series was written in translation by Postgate in the form of " musical sketches " or graphs that he drew for Elliott, who would then convert the drawings into a musical score.

The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above.

* Graphics capability was not taken seriously in the original IBM design brief ; graphics were considered only from the perspective of generating static business graphics such as charts and graphs.

They were able to construct exactly 656 ( 5, 5, 42 ) graphs, arriving at the same set of graphs through different routes.

There were five typewriters for printed output, five paper tape punches, and four pen plotters to produce graphs.

Another research branch continues the work on existential graphs of Charles Sanders Peirce, which were one of the origins of conceptual graphs as proposed by Sowa.

Random graphs were first defined by Paul Erd &# 337 ; s and Alfréd Rényi in their 1959 paper " On Random Graphs " and independently by Gilbert in his paper " Random graphs ".

Turán graphs were first described and studied by Hungarian mathematician Paul Turán in 1941, though a special case of the theorem was stated earlier by Mantel in 1907.

Scene graphs were generally left to the developer to implement, and it was all too common to see poor examples that led to poor performance.

In transcribing to modern notation, where no compound graphs as ligatures exist, editors usually indicate by a hook, a bracket / brace, or ( less often in polyphonic music ) a slur / phrase mark those notes that were combined into a ligature.

Such graphs are termed indicator diagrams and were first used by James Watt and others to monitor the efficiency of engines.

* A computational reasoning approach, where the relationships between graphs and probabilities were formally introduced.

Cang Jie, scribe for the Yellow Emperor, on looking at the tracks of the feet of birds and animals, realizing that the patterns and forms were distinguishable, started to create graphs, so that all kinds of professions could be regulated, and all people could be kept under scrutiny .( tr.

Even as copyists transcribed the main text of the book in clerical script in the late Han, and then in modern standard script in the centuries to follow, the small seal characters continued to be copied in their own ( seal ) script to preserve their structure, as were two kinds of variant graphs included by Xu, which he termed ancient script ( gǔwén 古文 ) and Zhòu script ( Zhòuwén 籀文, not to be confused with the Zhou Dynasty ).

Since the ease of writing with a brush is even greater than that of writing with a stylus in wet clay, it is assumed that the style and structure of Shang graphs on bamboo were similar to those on bronzes, and also that the majority of writing occurred with a brush on such books.

The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps.

* If Peggy is asked to show that the two graphs are isomorphic, she first uncovers all of H ( e. g. by turning all pieces of papers that she put on the table ) and then provides the vertex translations that map G to H. Victor can verify that they are indeed isomorphic.

Marginal costs are often shown on these graphs, with marginal cost representing the cost of the last unit produced at each point ; marginal costs are the slope of the cost curve or the first derivative of total or variable costs.

Perhaps the first instance of an axiom or rule with the power of C2 was the " Rule of ( De ) Iteration ," combining T13 and AA = A, of C. S. Peirce's existential graphs.

The first use of the phrase " perfect graph " appears to be in a 1963 paper of Claude Berge, after whom Berge graphs are named.

Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in linear time, it is possible to recognize chordal graphs in linear time.

In his 1961 and 1963 works defining for the first time this class of graphs, Claude Berge observed that it is impossible for a perfect graph to contain an odd hole, an induced subgraph in the form of an odd-length cycle graph of length five or more, because odd holes have clique number two and chromatic number three.

Kurzweil supports this final postulate with logarithmic graphs of the chronology of important events in the history of the Universe ( i. e. -the Big Bang, the origin of life, the birth of the human race, the creation of the first computer ).

The version of the linking number used for defining linkless embeddings of graphs is found by projecting the embedding onto the plane and counting the number of crossings of the projected embedding in which the first curve passes over the second one, modulo 2.

The graphs with Colin de Verdière graph invariant at most μ, for any fixed constant μ, form a minor-closed family, and the first few of these are well-known: the graphs with μ ≤ 1 are the linear forests ( disjoint unions of paths ), the graphs with μ ≤ 2 are the outerplanar graphs, and the graphs with μ ≤ 3 are the planar graphs.

In 1786, William Playfair published the first data graphs in his book The Commercial and Political Atlas.

These were also the first mathematic graphs.

The first step of defuzzification typically " chops off " parts of the graphs to form trapezoids ( or other shapes if the initial shapes were not triangles ).

In contrast to real numbers that have the property of varying " smoothly ", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

The graphs studied in graph theory should not be confused with the graphs of functions or other kinds of graphs.

The graph enumeration problem of counting directed acyclic graphs was studied by.

Erdős and Rényi ( 1960 ) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them.

Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.

In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry which satisfy the axioms of a building, but may not be connected with any group.

Isoperimetric profiles have been studied for Cayley graphs of discrete groups and for special classes of Riemannian manifolds ( where usually only regions A with regular boundary are considered ).

In contrast to real numbers that have the property of varying " smoothly ", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.

Most commonly studied is the one proposed by Edgar Gilbert, denoted G ( n, p ), in which every possible edge occurs independently with probability p. A closely related model, the Erdős – Rényi model denoted G ( n, M ), assigns equal probability to all graphs with exactly M edges.

Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied.

studied the analogue of the 15 puzzle on arbitrary finite connected and non-separable graphs.

Most, if not all, of the objects studied in finite mathematics are countable sets, such as integers, finite graphs, and formal languages.

Several researchers have studied the complexity of exponential time algorithms restricted to cubic graphs.

The processes studied include those in the theories of scheduling, bin packing, sequential selection, graphs, and dynamic allocation, along with those in queueing, polling, reservation, moving-server, networking, and distributed local-rule systems ( e. g. cellular automata ).

In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.

One property studied on graphs with two types of individuals is the fixation probability, which is defined as the probability that a single, randomly placed mutant of type A will replace a population of type B.

Generalizations have been studied by D. G. Higman ( coherent configurations ) and B. Weisfeiler ( distance regular graphs ).

Semi-symmetric graphs were first studied by Jon Folkman in 1967, who discovered the smallest semi-symmetric graph, the Folkman graph on 20 vertices.

Two-graphs have been studied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and also finite groups because many regular two-graphs have interesting automorphism groups.