11 pennies, forming a penny graph with 11 vertices and 19 edgesThe Hanoi graph as a penny graph (the contact graph of the black disks)
In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for every pair of tangent circles.[1] The circles can be represented physically by pennies, arranged without overlapping on a flat surface, with a vertex for each penny and an edge for each two pennies that touch.
Penny graphs have also been called unit coin graphs,[2] because they are the coin graphs formed from unit circles.[1] If each vertex is represented by a point at the center of its circle, then two vertices will be adjacent if and only if their distance is the minimum distance among all pairs of vertices. Therefore, penny graphs have also been called minimum-distance graphs,[3]smallest-distance graphs,[4] or closest-pairs graphs.[5] Similarly, in a mutual nearest neighbor graph that links pairs of points in the plane that are each other's nearest neighbors, each connected component is a penny graph, although edges in different components may have different lengths.[6]
Every penny graph is a unit disk graph and a matchstick graph.
Like planar graphs more generally, they obey the four color theorem, but this theorem is easier to prove for penny graphs.
Testing whether a graph is a penny graph, or finding its maximum independent set, is NP-hard; however, both upper and lower bounds are known for the size of the maximum independent set, higher than the bounds that are possible for arbitrary planar graphs.
^ abCite error: The named reference links was invoked but never defined (see the help page).
^Cite error: The named reference cffp was invoked but never defined (see the help page).
^Cite error: The named reference csiz was invoked but never defined (see the help page).
^Cite error: The named reference bmp was invoked but never defined (see the help page).
^Cite error: The named reference rcv was invoked but never defined (see the help page).
^Cite error: The named reference ew96 was invoked but never defined (see the help page).
In geometric graph theory, a pennygraph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other...
Eulerian and a pennygraph (this implies that it is unit distance and planar). It is also a 1-vertex-connected graph and a 2-edge-connected graph. There are...
distance graphs include the cactus graphs, the matchstick graphs and pennygraphs, and the hypercube graphs. The generalized Petersen graphs are non-strict...
any squaregraph as a matchstick graph. Every matchstick graph is a unit distance graph. Pennygraphs are the graphs that can be represented by tangencies...
graph can be represented as a contact graph of circles. The contact graphs of unit circles are called pennygraphs. Representations as contact graphs...
cycles in unit disk graphs Indifference graph, a one-dimensional analogue of the unit disk graphsPennygraph, the unit disk graphs for which the disks...
H_{3}^{n}} . These graphs have 3n vertices (OEIS: A000244) and 3(3n − 1)/2 edges (OEIS: A029858). They are pennygraphs (the contact graphs of non-overlapping...
creations full-time. Originally, like many webcomics, Penny Arcade was supported solely by donations. A graph on the main page indicated how much people had...
graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs...
Penny Dreadful is a British-American horror drama television series created and written by John Logan, who serves as executive producer alongside Sam...
Matching pennies is a non-cooperative game studied in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly...
of Waterloo. Her research interests include extremal combinatorics and graph theory. Haxell earned a bachelor's degree in 1988 from the University of...
Microsoft Graph is a Microsoft API developer platform that connects multiple services and devices. Initially released in November 2015 as Office 365 Unified...
Archived from the original on January 3, 2013. Retrieved December 27, 2012. "Penny Marshall's Ex-Boyfriend Art Garfunkel Once Credited Her for Pulling Him...
television film The Operation (1990), a crime, drama, TV movie starring Joe Penny, Lisa Hartman, and Jason Beghe The Operation (1992–1998), a reality television...
three neighbors. A free tree or unrooted tree is a connected undirected graph with no cycles. The vertices with one neighbor are the leaves of the tree...
oblique. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x...
analytics Inspired by Facebook's "social graph", LinkedIn CEO Jeff Weiner set a goal in 2012 to create an "economic graph" within a decade. The goal was to create...
known as square rectangular cuboid. They can be represented as the prism graph Π 4 {\displaystyle \Pi _{4}} . In the case that all six faces are squares...
mathematician, known for his work in combinatorics and graph theory. Bondy received his Ph.D. in graph theory from the University of Oxford in 1969. His advisor...
boundaries at the next election. The Conservative Party's candidate is Penny Mordaunt, the Leader of the House of Commons. Richmond and Northallerton...
Spataro, the head of Microsoft 365, Copilot for Microsoft 365 uses Microsoft Graph, an API, to evaluate context and available Microsoft 365 user data before...
Control and Prevention. Click on "Rising Rates" tab for a graph. See data table below the graph. Nelson, Lewis H.; Flomenbaum, Neal; Goldfrank, Lewis R...
issuance of other coins, which have values ranging from one cent (U.S. Penny) to 100 dollars. These other coins are more fully described in Coins of...
cover of a graph. The goal here is to find the minimum vertex cover of a graph: a collection of vertices such that each edge in the graph contains at...